j. differential geometry 60 (2002) 199-293

A DEGENERATION FORMULA OF GW-INVARIANTS

JUN LI

Abstract

This is the sequel to the paper [23]. In this paper, we construct the virtual moduli cycles of the degeneration of the moduli of stable morphisms constructed in [23]. We also construct the virtual moduli cycles of the moduli of relative stable morphisms of a pair of a smooth divisor in a smooth variety. Based on these, we prove a degeneration formula of the Gromov-Witten invariants.

0. Introduction

This is the second part of the project initiated in [23].

Like Donaldson invariants of 4-manifolds, Gromov-Witten invariants are intersection theories on the moduli spaces of stable morphisms to varieties or symplectic manifolds. Unlike the Donaldson invariants, in this case one needs to use virtual intersection theories to define the Gromov-Witten invariants: namely, the intersection theories defined via virtual moduli cycles. Such cycles were first constructed by Tian and the author [25, 26] for algebraic varieties, and an alternative construction was achieved by Behrend-Fantechi [2, 3]. Gromov-Witten invariants of general symplectic manifolds were developed in [32, 31, 6, 27, 33, 34] and the equivalence of these constructions were proved in [28, 35, 19].

The goal of this project is to prove a degeneration formula of the Gromov-Witten invariants in algebraic geometry. This is the analogy of the Donaldson-Floer theory in gauge theory. Such a degeneration theory (for Gromov-Witten theory) was investigated by several groups using analysis [5, 14, 15, 24, 37].

Supported partially by NSF grants.

Received November 2, 2001.

Here is the situation we will study in this paper. Let W → C be a family of projective schemes over a smooth pointed curve 0 ∈ C so that the total space W is smooth, the fibers W_{t} over t ≠ 0 ∈ C are smooth varieties and the special fiber W_{0} has two smooth irreducible components Y_{1} and Y_{2} intersecting transversally along a connected smooth divisor D ∈ W_{0}. We will call Y_{i}^{rel} ≜ (Y_{i}, D_{i}), where D_{i} = D ⊂ Y_{i}, the relative pairs after decomposing W_{0}. In this paper, we will construct the Gromov-Witten invariants of W_{t} for all t ∈ C. We will construct the relative Gromov-Witten invariants of the relative pairs Y_{i}^{rel} ≜ (Z, D). We will prove that the Gromov-Witten invariants of W_{t} are locally constant for t ∈ C. Finally we will prove a degeneration formula relating the Gromov-Witten invariants of W_{0} with the relative Gromov-Witten invariants of Y_{1}^{rel} and Y_{2}^{rel}, one in cycle form and the other in numerical form.

The first part of this paper is devoted to constructing the Gromov-Witten invariants of the singular scheme W_{0}. This is based on the moduli of relative stable morphisms to W/C, constructed in the first part of this project [23]. Recall that there we constructed a stack 𝔚 that includes all expanded degenerations of W. We then introduced the notion of prestable, predeformable and stable morphisms to 𝔚. We let Γ be the triple consisting of the genus, the number of marked points and the degree of the stable morphisms. We then constructed the moduli space 𝔐(𝔚, Γ) of stable morphisms to 𝔚 of topological type Γ and proved that it is a Deligne-Mumford stack, separated and proper over C. Applying parallel construction to a relative pair Z^{rel} (of a smooth divisor D in a smooth variety Z) we constructed the stack of expanded relative pairs ℨ^{rel}. We then defined the notion of relative stable morphisms to ℨ^{rel} and showed that the moduli 𝔐(ℨ^{rel}, Γ′) of relative stable morphisms to ℨ^{rel} with topological type Γ′ is also a separated and proper Deligne-Mumford stack. Here Γ′ is the topological type of the relative stable morphisms, to be explained momentarily.

In this paper, we first construct the standard obstruction theory of 𝔐(𝔚, Γ), 𝔐(𝔚_{0}, Γ) = 𝔐(𝔚, Γ) ×_{C} 0 and 𝔐(ℨ^{rel}, Γ′). We then show that they are all perfect, thus allow us to construct their respective virtual moduli cycles. Based on the virtual moduli cycle [𝔐(𝔚_{0}, Γ)]^{virt}, we define the Gromov-Witten invariants of W_{0}

Ψ_{Γ}^{W_{0}} : H^{∗}(W_{0})^{×k} × H^{∗}(𝔐_{g,k}) ⟶ ℚ

in the standard way, where Γ = (g, k, A). Similarly, for Z^{rel} we define its relative Gromov-Witten invariants

Ψ_{Γ′}^{Z^{rel}} : H^{∗}(Z)^{×k} × H^{∗}(𝔐_{Γ^{o}}) ⟶ H_{∗}(D^{r})

to be

Ψ_{Γ′}^{Z^{rel}}(α, β) = q_{∗}(ev^{∗}(α) ∪ π_{Γ′}^{∗}(β) [𝔐(ℨ^{rel}, Γ′)]^{virt}) ∈ H_{∗}(D^{r}).

Here k (resp. r) is the number of ordinary (resp. distinguished^{1} ) marked points of the domain curves, ev is the evaluation morphism associated to the ordinary marked points, q : 𝔐(ℨ^{rel}, Γ′) → D^{r} is the evaluation associated to the distinguished marked points and π_{Γ} : 𝔐(ℨ^{rel}, Γ′) → 𝔐_{Γ^{o}} is the forgetful morphism, where 𝔐_{Γ^{o}} is the moduli of stable nodal curves^{2} whose topology is given by the data in Γ′.

The invariants Ψ_{Γ}^{W_{0}} and Ψ_{Γ′}^{Z^{rel}} have the expected properties. For instance, Ψ_{Γ}^{W_{t}} is locally constant for t ∈ C, and Ψ_{Γ′}^{Z^{rel}} is invariant under any smooth deformation of Z^{rel}.

The second part of this paper is to derive a degeneration formula of the Gromov-Witten invariants associated to the (degeneration) family W. As explained in [23], we expect to have a formula relating the Gromov-Witten invariants of W_{0}, and hence of W_{t}, to the relative Gromov-Witten invariants of Y_{i}^{rel}. In this paper, we prove such a degeneration formula:

Ψ_{Γ}^{W_{t}}(α(t), β)

= ∑_{η∈Ω/∼} \frac{m(η)}{| Eq(η)|} ∑_{j∈K_{η}} [Ψ_{Γ_{1}}^{Y_{1}^{rel}}(ȷ_{1}^{∗}α(0), β_{η,1,j}) • Ψ_{Γ_{2}}^{Y_{2}^{rel}}(ȷ_{2}^{∗}α(0), β_{η,2,j})]_{0} .

We will explain the notation momentarily. We call the above the degeneration formula in numerical form. There is a parallel degeneration formula in cycle form:

[𝔐(𝔚_{0}, Γ)]^{virt}

= ∑_{η∈Ω/∼} \frac{m(η)}{| Eq(η)|} Φ_{η∗}Δ^{!}([𝔐(𝔜_{1}^{rel}, Γ_{1})]^{virt} × [𝔐(𝔜_{2}^{rel}, Γ_{2})]^{virt}).

The degeneration formula in numerical form is an easy consequence of the degeneration formula in cycle form.

1The distinguished marked points of the domain of a relative stable morphism f are mapped to the distinguished divisor D ⊂ Z under f.

2By which we mean nodal curves having no vector fields, not necessary connected.

We now explain the notation in the degeneration formulas. Let Γ_{1} be a topological type of a relative stable morphism f to Y_{1}^{rel}. As introduced in [23], Γ_{1} is a weighted graph that consists of vertices, ordered legs and ordered roots. Each leg or root is a line segment with one end attached to a vertex. Each vertex represents a connected component of the domain X of f (not necessarily connected), each leg (resp. root) represents a marked point (resp. distinguished marked point) on X. We require all distinguished marked points are mapped to D under f. The vertices (resp. roots) of Γ are assigned weights, representing the degrees of f along the associated components of X (resp. the contact orders of f along the normal direction of D ⊂ Z).

We define the evaluation morphism

q_{1} : 𝔐(𝔜_{1}^{rel}, Γ_{1}) ⟶ D^{r}; f ↦ (f(q_{1}), …, f(q_{r}))

by evaluating on the distinguished marked points (q_{1}, …, q_{r}) of X. Now let Γ_{1} and Γ_{2} be so that they have identical number of roots. Then we have a pair of evaluation morphisms q_{1} and q_{2} and thus can form the Cartesian product

𝔐(𝔜_{1}^{rel}, Γ_{1}) ×_{D^{r}} 𝔐(𝔜_{2}^{rel}, Γ_{2}).

Let Δ : D^{r} → D^{r} × D^{r} be the diagonal morphism. The virtual moduli cycle of the above Cartesian product is

Δ^{!}([𝔐(𝔜_{1}^{rel}, Γ_{1})]^{virt} × [𝔐(𝔜_{2}^{rel}, Γ_{2})]^{virt}).

The set Ω (in the summation above) is the set of all admissible triples (Γ_{1}, Γ_{2}, I). Here (Γ_{1}, Γ_{2}, I) is admissible if Γ_{1} and Γ_{2} are two weighted graphs associated to two relative stable morphisms that satisfy the following property: First their ordered sets of weighted roots are isomorphic. Hence if (f_{1}, f_{2}) is an element in 𝔐(𝔜_{1}^{rel}, Γ_{1})×_{D^{r}} 𝔐(𝔜_{2}^{rel}, Γ_{2}) with X_{i} the domain of f_{i}, then we can glue the i-th distinguished marked points (associated to the i-th root) of X_{1} with the i-th distinguished marked point of X_{2} for all i to obtain a new nodal curve X and a morphism f : X → W_{0}. As part of the requirement, the curve X is connected of arithmetic genus g and f is a stable morphism of degree d. Lastly, I is a rule assigning an ordering of the union of the ordinary marked points of X_{1} and X_{2}, consistent with their original orderings. Now let η = (Γ_{1}, Γ_{2}, I) be an admissible triple with r roots. Then any permutation σ ∈ S_{r} defines a new element η^{σ} by reordering the roots of η according to σ. For η_{1}, η_{2} ∈ Ω, we say η_{1} ∼ η_{2} if η_{1} = η_{2}^{σ} for some σ. The notation Eq(η) appeared in the degeneration formula is the set of all σ ∈ S_{r} so that η = η^{σ}.

Let η ∈ Ω/∼, then the above construction associates to every pair (f_{1}, f_{2}) a stable morphism f in 𝔐(𝔚, Γ). This defines a morphism

Φ_{η} : 𝔐(𝔜_{1}^{rel}, Γ_{1}) ×_{D^{r}} 𝔐(𝔜_{2}^{rel}, Γ_{2}) ⟶ 𝔐(𝔚_{0}, Γ).

A lemma in [23] asserts that Φ_{η} is a local immersion and the degree of Φ_{η}, as morphism to Im(Φ_{η}), is | Eq(η)|. Then the degeneration formula in cycle form asserts that the union of

\frac{𝔪(η)}{| Eq(η)|}Φ_{η∗}Δ^{!}([𝔐(𝔜_{1}^{rel}, Γ_{1})]^{virt} × [𝔐(𝔜_{2}^{rel}, Γ_{2})]^{virt}

is the virtual moduli cycle of 𝔐(𝔚_{0}, Γ). Here 𝔪(η) is the product of the weights of the roots of η (or of the Γ_{1} in η).

We now explain briefly the strategy to prove the degeneration formula. Let (1, t) be the pair of the trivial line bundle on 𝔐(𝔚, Γ) and the pullback of a section t ∈ Γ(O_{C}) so that t^{−1}(0) is the origin 0 ∈ C. Then the virtual moduli cycle [𝔐(𝔚_{0}, Γ)]^{virt} is the intersection

[𝔐(𝔚, Γ)]^{virt} ∩ t^{−1}(0) ∈ A_{∗}𝔐(𝔚_{0}, Γ).

Using the notion of localized top Chern class, this is c_{1}(1,t)[𝔐(𝔚, Γ)]^{virt}.

It turns out that to each η ∈ Ω there associates a pair (L_{η}, t_{η}) of a line bundle L_{η} on 𝔐(𝔚, Γ) and a section t_{η} ∈ Γ(L_{η}) so that 1 ≅ ⊗_{η∈Ω/∼}L_{η} as line bundles and under this isomorphism t = Π_{η∈Ω/∼}t_{η}. This says that 𝔐(𝔚_{0}, Γ) “virtually” is a union of normal crossing divisors, each associated to an η ∈ Ω and is defined by the vanishing of t_{η}. This can be seen as follows: Let f ∈ 𝔐(𝔚_{0}, Γ) be a general point, say represented by f:X → W_{0}. Then f_{1} = f|_{X_{1}} with X_{1} = f^{−1}(Y_{1}) defines a relative stable morphism to Y_{1}^{rel}. Similarly we have the induced relative stable morphism f_{2} : X_{2} → Y_{2}^{rel}. Let Γ_{1} and Γ_{2} be the topological types of f_{1} and f_{2}, respectively. The fact that f can be reconstructed from the pair (f_{1}, f_{2}) provides us a triple η = (Γ_{1}, Γ_{2}, I), which belongs to Ω. The general points f ∈ 𝔐(𝔚_{0}, Γ) that share identical η defines a closed subset in 𝔐(𝔚_{0}, Γ). This set is homeomorphic to t_{η}^{−1}(0). The miracle is that such closed subset carries a natural closed subscheme structure, and is in fact defined by the vanishing of a Cartier divisor (L_{η}, t_{η}). Further the tensor product of all such (L_{η}, t_{η}) is the (1, t) that defines the moduli stack 𝔐(𝔚_{0}, Γ). Then by applying the known identity of the localized top Chern class, we have

[𝔐(𝔚_{0}, Γ)]^{virt} = c_{1}(1, t)[𝔐(𝔚, Γ)]^{virt} = ∑_{η∈Ω/∼} c_{1}(L_{η}, t_{η})[𝔐(𝔚, Γ)]^{virt}.

To prove the degeneration formula in cycle form, we need to show that

c_{1}(L_{η}, t_{η})[𝔐(𝔚, Γ)]^{virt}

= \frac{𝔪(η)}{| Eq(η)|}Φ_{η∗}Δ^{!}([𝔐(𝔜_{1}^{rel}, Γ_{1})]^{virt} × [𝔐(𝔜_{2}^{rel}, Γ_{2})]^{virt}).

This is done as follows: First we will show that the vanishing locus t_{η}^{−1}(0) ⊂ 𝔐(𝔚, Γ) is homeomorphic to the image stack of Φ_{η}. We then show that the cycle c_{1}(L_{η}, t_{η})[𝔐(𝔚, Γ)]^{virt} is a multiple of the virtual moduli cycle of 𝔐(𝔜_{1}^{rel}, Γ_{1}) ×_{D^{r}} 𝔐(𝔜_{2}^{rel}, Γ_{2}), endowed with the obstruction theory induced by the Cartesian product. This leads to the formula above.

Finally, we explain the notations in the degeneration formula. This formula is an immediate corollary of the degeneration formula in cycle form. The only new symbols are ȷ_{i∗}, K_{η} and β_{η,i,j}. First ȷ_{i}: Y_{i} → W_{0} is the inclusion and hence ȷ_{i}^{∗}α(0) is the pullback cohomology. Secondly, given any η = (Γ_{1}, Γ_{2}, I) ∈ Ω, we can form the moduli space of stable curves (not necessary connected but with no vector fields) whose topology are determined by Γ_{i}. We denote such moduli space by 𝔐_{Γ_{i}^{o}}. For any pair (C_{1}, C_{2}) ∈ 𝔐_{Γ_{1}^{o}} × 𝔐_{Γ_{2}^{0}} we can glue C_{1} with C_{2} pairwise along all pairs of the i-th distinguished nodes. This defines a morphism G_{η} : 𝔐_{Γ_{1}^{0}} × 𝔐_{Γ_{2}^{o}} ⟶ 𝔐_{g,k}. The terms β_{η,i,j} are terms appear in the Kunneth decomposition,

G_{η}^{∗}(β) = ∑_{j∈K_{η}} β_{η,1,j} ⊠ β_{η,2,j},

assuming it exists.

As mentioned in the introduction of [23], the construction of the moduli stacks 𝔐(𝔚, Γ) and 𝔐(ℨ^{rel}, Γ), and the derivation of the degeneration formula in this paper will be useful in studying several problems in algebraic geometry, some related to mathematical physics. Some of these will be addressed in the future research.

The degeneration formula of Gromov-Witten invariants for symplectic manifolds has been pursued by several groups prior this work.

(In symplectic setting this is when a smooth symplectic manifold degenerates to a union of two smooth symplectic manifolds intersecting transversally, called the symplectic sum.) In [37], Tian studied the symplectic sums for semi-positive symplectic manifolds and showed how to derive the decomposition formula of the Gromov-Witten invariants in this setting. Later, A. Li-Ruan [24] worked out a version of the relative Gromov-Witten invariants and the degeneration formula of Gromov-Witten invariants for general symplectic manifolds and symplectic sum. The degeneration formula in numerical form proved in this paper is analogous to the degeneration formula in [24]. A parallel theory was developed by Ionel-Parker around the same time [14, 15, 16]. Their formula works for more general cases and is largely analogous to ours. It contains a correction term, which is expected to be trivial when the symplectic sum is along a (holomorphic) divisor. The SFT theory of Eliashberg-Givental-Hofer [5] is a very general theory part of which can be interpreted as research along this line. The degeneration formula in cycle form proved here is new.

This paper consists of five sections. In Section 1, we work out the obstruction theory of 𝔐(𝔚, Γ), 𝔐(ℨ^{rel}, Γ) and other related moduli stacks. The main result of this section is that the standard obstruction theories of these moduli stacks are perfect. Section 2 is devoted to construct the virtual moduli cycles of these moduli stacks. We present a modified construction of virtual moduli cycles which allow one to construct such cycles without assuming the existence of global locally free sheaves that resolve the obstruction sheaves, as assumed in [26] (and also assumed in [3] but recently removed in [22]). The Gromov-Witten invariants of W_{0} and the relative Gromov-Witten invariants of Z^{rel} are constructed in this section. In Section 3, we constructed the line bundles with sections (L_{η}, t_{η}) mentioned in the introduction and in Section 4 we demonstrate how to derive the degeneration formula, assuming a series of key lemmas. The last section is devoted to the proof of these key lemmas. In the Appendix, we give an expression of the obstruction space of a closed point in 𝔐(𝔚, Γ) and in 𝔐(ℨ^{rel}, Γ) in terms of some known cohomology groups.

1. Deformation theory of log morphisms

In this section we will first recall the notion of morphisms between schemes with log structures (in short log morphism). We will then show that the notion of predeformable morphisms introduced in [23] is a special class of log morphisms. In the end, we will work out the deformation theory of log morphisms.

1.1 Predeformable morphisms and log morphisms

Let f : X → W [n] be a flat family of predeformable morphisms over S with g : S → A^{n+1} the morphism underlying f (cf. [23, §2]). In this subsection, we will give X and W [n] canonical log structures and show that f is a morphism between schemes with log structures. We will then use the sheaf of log differentials to describe the deformation of predeformable morphisms. All materials concerning schemes with log structures are drawn from the papers of Kato [17, 18].

We first recall the notion of logarithmic structures (log structures) on schemes, following [17, 18]. Let X be any scheme with O_{X} its structure sheaf. We view O_{X} as a sheaf of monoids under multiplication.

Definition 1.1. A prelog structure on X is a homomorphism α: M → O_{X} of sheaves of monoids, where M is a sheaf of commutative monoids on the étale site X_{et} of X. The prelog structure α: M → O_{X} is said to be a log structure if α induces an isomorphism α:α^{−1}(O_{X}^{×}) → O_{X}^{×}, where O_{X}^{×} is the subsheaf of invertible elements in O_{X}.

Given a prelog structure, one can construct canonically an associated log structure α^{a}:M^{a} → O_{X}, where M^{a} = (M⊕O_{X}^{×})/α^{−1}(O_{X}^{×}) (cf. [18, §2]). A morphism (X, M) → (Y, N) of schemes with log structures is a pair (f, h) of a morphism f:X → Y and a homomorphism h:f^{−1}(N) → M that satisfy the obvious commutativity condition: The composite f^{−1}(N) → M → O_{X} is identical to f^{−1}(N) → f^{−1}(O_{Y}) → O_{X}. For convenience, given a scheme (X, M) with local log structure, we shall abbreviate it to X^{†} with the local log structure M implicitly understood. Accordingly, we will abbreviate a morphism (X, M) → (Y, N) between schemes with log structures by f:X^{†} → Y^{†}.

A typical example of a scheme with log structure is the log structure of a pair (X, D) of a smooth scheme X and a divisor D ⊂ X with normal crossing singularities (cf. [17, (1.5)]). Let

π_{n}:W [n] → A^{n+1}

be the expanded family constructed in [23, §2]. Recall that W [n] is a small resolution of W ×_{C} C[n] constructed in [23]^{3} . Then W [n]×_{A^{1}} 0 ⊂ W [n] is a divisor with normal crossing singularities having n + 2 irreducible components. Here A^{n+1} → A^{1} is the morphism (z_{1}, ..., z_{n+1}) ↦ z_{1} … z_{n+1}. We let W [n]^{†} and A^{n+1†} be W [n] and A^{n+1} endowed with the log structures induced by their respective pairs

W [n] ×_{A^{1}} 0 ⊂ W [n] and A^{n+1} ×_{A^{1}} 0 ⊂ A^{n+1},

Let X^{†} be a log scheme. A chart of X^{†} consists of an étale neighborhood U ⊂ X, a constant sheaf of monoids P and a homomorphism P → O_{U} so that the associated log structure P^{a} of P is isomorphic to the log structure on U. We now describe the charts of W [n]^{†}. Recall that the projection π_{n} is smooth away from n + 1 disjoint smooth codimension 2 subvarieties D_{1}, …, D_{n+1}, indexed so that the component D_{l} surjects onto the l-th coordinate hyperplane H_{l} ⊂ A^{n+1}. Let y ∈ D_{l} be any point. A chart of W [n] along y consists of a pair (W, ψ), where W is a Zariski open subset of y ∈ W [n] and ψ is a smooth morphism

ψ : W ⟶ Spec 𝕜[w_{1}, w_{2}] ⊗_{𝕜[t_{l}]} Γ(A^{n+1}) ≜ Θ_{l} (1.1)

so that the canonical projections W → A^{n+1} is the composite of ψ with the projection Θ_{l} → A^{n+1}. Here the two projections in the fiber product are defined via t_{l} ↦ w_{1}w_{2} and by viewing t_{l} as the l-th standard coordinate variable of A^{n+1}. Let ℕ^{2} → O_{W} be the homomorphism of monoids^{4} defined by e_{i} ↦ ψ^{∗}(w_{i}) and let ℕ^{n+1} → O_{A^{n+1}} be defined via e_{l} ↦ t_{l}. We then form the product ℕ^{2} ×_{ℕ_{l}} ℕ^{n+1} where ℕ_{l} ≡ ℕ, ℕ_{l} → ℕ^{2} is defined by e ↦ e_{1} + e_{2} while ℕ_{l} → ℕ^{n+1} is the inclusion as the l-th copy in ℕ^{n+1}. Because of the relation t_{l} ↦ w_{1}w_{2}, the homomorphism

ℕ^{2} ×_{ℕ_{l}} ℕ^{n+1} ⟶ O_{W} (1.2)

induced by ℕ^{2} → O_{W} and ℕ^{n+1} → O_{A^{n+1}} → O_{W} defines a prelogarithmic structure on W. This defines a chart of W [n]^{†} near y ∈ D_{l}. Similarly, the log structure of A^{n+1} is given by the homomorphism ℕ^{n+1} → O_{A^{n+1}} via e_{l} ↦ t_{l}.

3We first fix an étale C → A^{1} so that 0 ∈ C is the only point over 0 ∈ A^{1}. We then form C[n] = C ×_{A^{1}} A^{n+1}, where A^{n+1} → A^{1} is defined via (t_{i}) ↦ t = Πt_{i}. The family W [n]/C[n] has the property that it has smooth total space, its fibers have only normal crossing singularities and the fibers of W [n] are either W_{t} for t ≠ 0 ∈ C or unions of Y_{1}, Y_{2} and copies of a ruled variety Δ. Finally the G[n] = GL(1)^{×n} action on A^{n+1} via (t_{i})^{σ} = (σ_{1}t_{1}, σ_{1}^{−1}σ_{2}t_{2}, …,) lifts to a unique action on W [n].

4We will denote by ℕ the additive monoid of nonnegative integers, and by ℕ^{k} the direct product monoid with standard generators e_{1}, …, e_{k}.

Now let f: X → W [n] be a flat family of predeformable morphisms over S, defined in [23, §2]. Out next step is to show that we can endow X /S with a log structure so that f becomes a morphism between schemes with log structures. We will use the following convention throughout this paper. We call p:U → V an étale neighborhood of the family X /S if U and V are étale neighborhoods of X and S, respectively, such that the diagram

\begin{matrix}U & ⟶ & X \\ ^{p}↓ & & ^{π}↓ \\ V & ⟶ & S \end{matrix} (1.3)

is commutative.

Definition 1.2. Let π:X → S be a flat family of nodal curves and let x ∈ X be a node of the fibers of the family X /S. A chart of the nodes of X /S near x consists of an affine étale neighborhood p:U → V of X /S near x, two regular functions z_{1} and z_{2} ∈ Γ(O_{U}) and a regular function s ∈ Γ(O_{V}) satisfying z_{1}z_{2} = p^{∗}(s) such that the homomorphism

ϕ : 𝕜[z_{1}, z_{2}] ⊗_{𝕜[s]} Γ(O_{V}) ⟶ Γ(O_{U}) (1.4)

is an étale homomorphism, that {z_{1} = z_{2} = 0} ⊂ U is connected and that the induced homomorphism

ϕ̂: (𝕜[z_{1}, z_{2}] ⊗_{𝕜[s]} Γ(O_{V}))ˆ ⟶ Γ(O_{U})ˆ

is an isomorphism.

A few remarks are in order. First ϕ is defined by viewing z_{1}, z_{2} and s on one hand as formal variables and on the other hand as regular functions. This should cause no confusion since their roles are clear from the context. Also, the homomorphism 𝕜[s] → 𝕜[z_{1}, z_{2}] is defined by s ↦ z_{1}z_{2}. As a convention, in this section we will use L̂ and ĥ (or Lˆ and hˆ ) to denote the I-adic completion of the ring L and the image of h ∈ L under the homomorphism L → L̂, assuming I = (z_{1}, z_{2}) is an ideal of L.

Lemma 1.3. Let π:X → S be a flat family of nodal curves and let x ∈ X be a node in X_{y}, where y = π(x). Then there exists a chart of the nodes of X /S near x.

Proof. The proof is straightforward and will be omitted. q.e.d.

We next recall the notion of morphisms of pure contact and predeformable morphisms. Let f : X → W [n] be a flat family of morphisms over S. We let ξ ∈ X be a node of the fibers of X /S so that f(ξ) ∈ D_{l} for some l. (Such nodes are called the distinguished nodes.) We let (U/V, ϕ) be a chart of the node ξ of X /S, as in Definition 1.2, and let ψ : W → W [n] be a chart of f(ξ) as in (1.1) so that f(U) ⊂ W. We let f_{U} = f|_{U} : U → W. As mentioned, we denote by Â the (z_{1}, z_{2})- adic (resp. (s)-adic) completion of A in case A is a 𝕜[z_{1}, z_{2}]-algebra (resp. 𝕜[s]-algebra).

Definition 1.4. Let the notation be as before. We say f is formally of pure contact of order m at ξ (or is predeformable at ξ) if there is an étale chart U/V of ξ ∈ X /S and a chart W of f(ξ) so that the induced homomorphism

f̂_{U}^{∗} : 𝕜[w_{1}, w_{2}] → (𝕜[z_{1}, z_{2}] ⊗_{𝕜[s]} Γ(O_{V}))ˆ

has the property that there are units ĥ_{1}, ĥ_{2} ∈ (𝕜[z_{1}, z_{2}] ⊗_{𝕜[s]} Γ(O_{V}))ˆ satisfying ĥ_{1}ĥ_{2} ∈ Γ̂(O_{V}) so that, possibly after exchanging w_{1} and w_{2}, f̂_{U}^{∗}(w_{i}) = ĥ_{i}z_{i}^{m}. We say f is of pure contact of order m if there are charts as before and units h_{1}, h_{2} ∈ 𝕜[z_{1}, z_{2}]⊗_{𝕜[s]}Γ(O_{V}) satisfying h_{1}h_{2} ∈ Γ(O_{V}) such that, possibly after exchanging w_{1} and w_{2}, f_{U}^{∗}(w_{i}) = z_{i}^{m}h_{i}.

The notion of pure contact of order m was defined in [23, Definition 2.3]. We have the following facts whose proof can be found in the Appendix.

Lemma 1.5. The notion of pure contact is independent of the choice of the charts of the nodes of U/V.

Lemma 1.6. Let the notation be as in Definition 1.4. Then f is of pure contact at ξ if and only if it is formally of pure contact at ξ.

Let f : X → W [n] be a predeformable morphism over S. We now define the induced log structure on X /S. We begin with charts of f. Let x ∈ X be any closed point so that f(x) ∉ D. Here D is the union of all D_{l}. A chart of f near x is a triple (U_{α}/V_{α}, W_{α}, f_{α}) of an étale neighborhood U_{α}/V_{α} of x ∈ X, a Zariski neighborhood W_{α} ⊂ W [n] − D and f_{α} = f|_{U_{α}} so that f_{α}(U) ⊂ W_{α}. Charts of this type will be called of the first kind. Next let x ∈ X be a distinguished node of X /S, namely f(x) ∈ D_{l_{α}} for some l_{α}. We pick a chart (W_{α}, ψ_{α}) of f(x) ∈ W [n] with ψ_{α} as in (1.1). Because f is predeformable, by Lemma 1.6 we can find a chart of the nodes of X /S near x, say given by (U_{α}/V_{α}, ϕ_{α}) as in Definition 1.2, so that U_{α} → X → W [n] lifts to f_{α}: U_{α} → W_{α}, and further there are units h_{α,1}, h_{α,2} ∈ Γ(O_{U_{α}})^{×}, elements g_{α,1}, …, g_{α,n+1} ∈ Γ(O_{V_{α}}) and an integer m_{α} so that

h_{α,1}h_{α,2} ∈ Γ(O_{V_{α}}), f_{α}^{∗}(w_{α,i}) = z_{α,i}^{m_{α}}h_{α,i} and f_{α}^{∗}(t_{j}) = g_{α,j} (1.5)

for i = 1, 2 and j = 1, …, n + 1.^{5} Here we comment that we will not distinguish w_{α,i} (resp. z_{α,i}) with ψ_{α}^{∗}(w_{α,i}) (resp. ϕ_{α}^{∗}(z_{α,i})). In other words, we will view w_{α,i} ∈ O_{W_{α}} (resp. z_{α,i} ∈ O_{U_{α}}). Similarly we will view t_{l} ∈ O_{W [n]} via the pullback O_{A^{n+1}} → O_{W [n]}. Note that since w_{α,1}w_{α,2} = t_{l_{α}} and z_{α,1}z_{α,2} = s_{α}, we must have

g_{α,l_{α}} = s_{α}^{m_{α}}(h_{α,1}h_{α,2}). (1.6)

We will call such triplet (U_{α}/V_{α}, W_{α}, f_{α}) with (ϕ_{α}, ψ_{α}) understood a chart of f of the second kind.

Simplification 1.7. In case (U_{α}/V_{α}, W_{α}, f_{α}) is a chart of the second kind, for simplicity we assume h_{α,1} ≡ h_{α,2} ≡ 1. This is possible possibly after an étale base change of U_{α}/V_{α}.

We now cover X by charts of f, say (U_{α}/V_{α}, W_{α}, f_{α}) indexed by α ∈ Λ, of the first or the second kinds satisfying the Simplification 1.7. Let α be a chart of the second kind with ϕ_{α} and ψ_{α} understood. We then let M_{α}^{0} = ℕ^{2} (resp. N_{α}^{0} = ℕ) and let M_{α}^{0} → O_{U_{α}} (resp. N_{α}^{0} → O_{V_{α}}) be the prelog structure defined by e_{α,i} ↦ z_{α,i} (resp. e_{α} ↦ s_{α}). Note that N_{α}^{0} → M_{α}^{0} defined by e_{α} ↦ e_{α,1} + e_{α,2} makes the projection U_{α} → V_{α} a morphism between schemes with (their respective associated) log structures. We now define the desired log structures on S. Let ξ ∈ S be any closed point and let X_{ξ} be the fiber of X over ξ with Σ = f^{−1}(D)∩X_{ξ} be the set of distinguished nodes of X_{ξ}. We let Λ_{0} be a collection of α ∈ Λ so that {U_{α}}_{α∈Λ_{0}} covers a neighborhood of X_{ξ} ⊂ X. We then let K_{l} be those α so that f^{−1}(D_{l}) ∩ U_{α} ≠ ∅. We let K be the union of K_{1}, …, K_{n+1}. By eliminating redundant α from Λ_{0} we can assume that each node of X_{ξ} ∩ f^{−1}(D_{l}) is covered by at most one α ∈ K_{l}. We then pick an étale neighborhood V of ξ ∈ S so that V → S factor through V_{α} → S for all α ∈ Λ_{0}. For each l ∈ [n + 1] we let \overline{N}_{l} = ⊕_{α∈K_{l}}N_{α}^{0} and let N_{l} be the quotient (monoid) of \overline{N}_{l} by the relations m_{α}e_{α} = m_{β}e_{β} for all pairs α, β ∈ K_{l}. In case K_{l} = ∅, we agree N_{l} = \overline{N}_{l} = ℕ. Note that for α ∈ K_{l} the homomorphism \overline{N}_{l} → O_{V_{α}} defined by e_{α} ↦ s_{α} and the pullback O_{V_{α}} → O_{V} descends to a homomorphism N_{l} → O_{V}, because of the relations (1.5) and (1.6). When K_{l} = ∅, we define N_{l} → O_{V} via e ↦ g^{∗}(t_{l}). We define the prelog structures N_{V} and \overline{N}_{V} on V to be the ones given by the direct sums

5By this we mean it is in the image of the pullback homomorphism Γ(O_{V_{α}}) → Γ(O_{U_{α}}).

N_{V} ≜ ⊕_{l=1}^{n+1}N_{l} ⟶ O_{V} and \overline{N}_{V} ≜ ⊕_{l=1}^{n+1}\overline{N}_{l} ⟶ O_{V}. (1.7)

We denote by N_{V} and \overline{N}_{V} the associated log structures on V.

We next define the desired prelog structure on U_{α}. By replacing U_{α} by U_{α} ×_{V_{α}} V, we can assume V_{α} = V. Accordingly we let N_{α} and \overline{N}_{α} → O_{V} be that induced by N_{V} and \overline{N}_{V}. Now let l be so that α ∈ K_{l}. Recall that U_{α} has a prelog structure given by M_{α}^{0} → O_{U_{α}}. Let N_{α}^{0} → M_{α}^{0} be as before (defined by e_{α} ↦ e_{α,1} + e_{α,2}). Then we have the obvious homomorphism

\overline{M}_{α} = M_{α}^{0} ×_{N_{α}^{0}} \overline{N}_{l} ⊕ (⊕_{l′≠l}\overline{N}_{l′}) ⟶ O_{U_{α}} (1.8)

and

M_{α} = M_{α}^{0} ×_{N_{α}^{0}} N_{l} ⊕ (⊕_{l′≠l}N_{l′}) ⟶ O_{U_{α}}. (1.9)

Here M_{α}^{0} ×_{N_{α}^{0}} \overline{N}_{l} → O_{U_{α}} is induced by M_{α}^{0} → O_{U_{α}} and \overline{N}_{l} → O_{V_{α}} → O_{U_{α}} while \overline{N}_{l′} → O_{U_{α}} is the composite of N_{l′} → O_{V_{α}} → O_{U_{α}}. The arrow in (1.9) is defined similarly. They define two prelog structures on U_{α}. We let \overline{M}_{α} → O_{U_{α}} and M_{α} → O_{U_{α}} be the associated log structures. Note that the obvious N_{α} → M_{α} and \overline{M}_{α} → \overline{N}_{α} make the projection U_{α} → V_{α} a morphism between schemes with respective log structures.

Proposition 1.8. The log structures (U_{α}, M_{α}) and (V_{α}, N_{α}) patch together to form log structures M on X and N on S. The same conclusion holds for (U_{α}, \overline{M}_{α}) and (V_{α}, \overline{N}_{α}). The collection of homomorphisms N_{α} → M_{α} makes X^{†} ≡ (X , M) a log scheme over S^{†} ≡ (S, N). Further, the morphism f is naturally a morphism between schemes with log structures X^{†}/S^{†} → W [n]^{†}/A^{n+1†}.

Proof. The fact that the so defined log structures on U_{α} and V_{α} patch together to form log structures on X and S is obvious. We now study the morphism f. We first investigate the morphism g:S → A^{n+1} underlying f. Let ξ ∈ S be any closed point with (V, N_{V}) a chart of S^{†}, constructed before. Recall that the log structure on A^{n+1†} is given by the prelog structure ℕ^{n+1} → Γ(A^{n+1}) via e_{l} ↦ t_{l}. To show g|_{V} : V → A^{n+1} is a morphism between schemes with log structures we need to define a homomorphism ℕ^{n+1} → N_{V} that satisfies the obvious compatibility condition. By definition N_{V} = ⊕_{l}N_{l} and N_{l} is ℕ in case K_{l} = ∅ and is the quotient of \overline{N}_{l} = ⊕_{α∈K_{l}}N_{α}^{0} otherwise. In the first case we define ℕ → N_{l} to be the unique isomorphism. In the later case we let ℕ → N_{l} be induced by 1 ↦ m_{α}e_{α} ∈ N_{l} for some α ∈ K_{l}. By the definition of N_{l} such definition is independent of α ∈ K_{l}. The desired homomorphism ℕ^{n+1} → N_{V} is the direct sum of these n + 1 copies ℕ → N_{l}. It is direct to check that this defines V → A^{n+1} a morphism between schemes with log structures. By working over a covering of S, one sees that g is a morphism between S^{†} → A^{n+1†}.

The proof that f a morphism between X^{†} and W [n]^{†} and is compatible to g: S^{†} → A^{n+1†} is similar, relying on the relations (1.5) and (1.6) and the assumption that all h_{α,i} ≡ 1. This completes the proof of the proposition. q.e.d.

We conclude this subsection by noting the equivalence of the deformations of predeformable morphisms and the deformations of log morphisms.

Definition 1.9.

1. A log extension of (V_{α}, N_{α}) by I consists of an extension Ṽ_{α} of V_{α} by I as schemes, and an extension N_{α} → OṼ_{α} of N_{α} → O_{V_{α}}. We denote such extension by (Ṽ_{α}, Ñ_{α}).

2. A flat log extension of (U_{α}/V_{α}, M_{α}/N_{α}) by I consists of an extension (Ṽ_{α}, Ñ_{α}) of (V_{α}, N_{α}) by I, a flat extension Ũ_{α} → Ṽ_{α} of U_{α} → V_{α} and an extension M_{α} → OŨ_{α} of the prelog structure M_{α} → O_{U_{α}} of which the following holds: a. The projection Ũ_{α} → Ṽ_{α} is a log morphism under the given N_{α} → M_{α}; b. Away from the distinguished nodes of U_{α} the prelog structure M_{α} → OŨ_{α} is the pullback of N_{α} → OṼ_{α}; c. Near the distinguished nodes in U_{α} the projection Ũ_{α} → Ṽ_{α} is log smooth.

We have the following lemma which says that extending f as a predeformable morphism is equivalent to extending f as a log morphism.

Lemma 1.10. Let f : X /S → W [n]/A^{n+1} be as before with the canonical log structures understood. Let S̃ ⊃ S be an extension of S. Suppose S̃^{†} is a log extension of S, X̃^{†} → S̃^{†} is a flat log extension of X /S and f̃: X̃^{†}/S̃^{†} → W [n]^{†}/A^{n+1†} is an extension of f as log morphism. Then f̃ is a predeformable extension of f and the log structures on X̃/S̃ induced by f̃ coincide with the log structure of X̃^{†}/S̃^{†}.

Proof. The proof is straightforward and will be omitted. q.e.d.

1.2 Deformation of predeformable morphisms

The goal of this subsection is to work out the deformation theory of predeformable morphisms. Based on the equivalence lemma, it is natural to work out the deformation theory of predeformable morphisms in the frame work of log morphisms. However, the deformation theory of log morphisms worked out in [17, 18] deal with the question on how to extend families over S^{†} to S̃^{†}. In our situation, the log structure on S for a predeformable family f : X /S → W [n]/A^{n+1} relies on the morphism f. Hence for any extension of base S ⊂ S̃ the extension of log structure to S̃ is part of the extension problem. Hence the deformation theory of Kato [18] can not be applied directly. In this subsection, we will work with the deformation of predeformable directly.

Our construction of the obstruction theory is elementary. We first show that locally there are no obstruction to extending predeformable morphisms. We then identify the space of all such deformations. After that, it is standard to express the obstruction to the deformation of (global) predeformable morphisms as a cohomology class in a cohomology of a complex built out of a Cěch complex. In the end, we check that this complex is a perfect complex.

We begin with recalling some basic notion in deformation theory. Our treatment follows [1, 26]. Let A be an \overline{A}-algebra. We let 𝔗𝔯𝔦_{A/\overline{A}} be the category whose objects consists of are all triples (B, I, φ) where B are \overline{A}-algebras, I are ideals of B such that I^{2} = 0 and φ are \overline{A}- homomorphisms A → B/I. Let ξ = (B, I, φ) and ξ′ = (B′, I′, φ′) be two objects in 𝔗𝔯𝔦_{A/\overline{A}}. A morphism from ξ to ξ′ consists of an \overline{A}- homomorphism r : B → B′ so that r(I) ⊂ I′ and φ′ = φ ∘ r_{0} where r_{0} : B/I → B′/I′ is the induced homomorphism. We let 𝔐𝔬𝔡_{A} be the category whose objects are pairs (B, I) where B are A-algebras and I are B-modules. Morphisms from (B, I) to (B′, I′) are pairs (r, r̃) where r: B → B′ are A-homomorphisms and r̃ are B-homomorphisms I → I′. We let 𝔐𝔬𝔡_{A}^{∗} be the category whose objects are (v, B, I) where (B, I) ∈ Ob(𝔐𝔬𝔡_{A}) and v ∈ I. Morphisms from (B, I, v) to (B′, I′, v′) are pairs (r, r̃) as in 𝔐𝔬𝔡_{A} so that r̃(v) = v′.

We define 𝔇𝔢𝔣_{A/\overline{A}}:𝔗𝔯𝔦_{A/\overline{A}} → (Sets) be the functor that associates to any ξ = (B, I, φ) the set of all \overline{A}-homomorphisms A → B extending φ: A → B/I. (In case \overline{A} is understood, we will omit the subscript \overline{A}.) It is known that under some mild conditions [36] this set admits B/I-module structures. Once this is the case, then after fixing a reference element a ∈ 𝔇𝔢𝔣_{A}(ξ) we can give 𝔇𝔢𝔣_{A}(ξ) a natural B/I-module structure. In particular, if B = B/I ∗ I, which is the trivial extension of B/I by the module I, then 𝔇𝔢𝔣_{A}(ξ) contains the trivial extension B → A induced by B = B/I ∗ I → B/I → A. Using this as the reference element, we obtain a canonical module structure on 𝔇𝔢𝔣_{A}(ξ). Note that there is a natural functor 𝔐𝔬𝔡_{A} → 𝔗𝔯𝔦_{A/\overline{A}} that sends (B, I) to (B ∗ I, I, φ) where φ:A → B is the obvious homomorphism. We let 𝔇𝔢𝔣_{A}^{1} : 𝔐𝔬𝔡_{A} → 𝔐𝔬𝔡_{A} be the composite of 𝔐𝔬𝔡_{A} → 𝔗𝔯𝔦_{A/\overline{A}} with 𝔇𝔢𝔣_{A}. We call 𝔇𝔢𝔣_{A}^{1} the functor of the first order deformations.

Now let E^{•} be any complex of A-modules. For any integer i the i-th cohomology of E^{•} defines a functor 𝔥^{i}(E^{•}) : 𝔐𝔬𝔡_{A} → 𝔐𝔬𝔡_{A} via (B, I) ↦ (B, h^{i}(E^{•} ⊗_{A} I)).

Definition 1.11. Let S = Spec A be an affine scheme over T = Spec \overline{A}. A perfect obstruction theory of S/T consists of a two term complex of finitely generated free A-modules E^{•} = [E^{1} → E^{2}] (indexed at [1, 2]) and an obstruction assignment 𝔬𝔟 taking value in the second cohomology of E^{•} of which the following hold:

1. The functor 𝔇𝔢𝔣^{1} is isomorphic to the functor 𝔥^{1}(E^{•}).

2. For any triple (B, I, φ) ∈ Ob(𝔗𝔯𝔦_{A/\overline{A}}), the element

𝔬𝔟(B, I, φ) ∈ 𝔥^{2}(E^{•})(I) = h^{2}(E^{•} ⊗_{A} I)

is the obstruction class to extending φ : A → B/I to an \overline{A}- homomorphism A → B.

3. The obstruction assignment

(B, I, φ) ↦ (B, h^{2}(E^{•} ⊗_{A} I), 𝔬𝔟(B, I, φ))

is a functor from 𝔗𝔯𝔦_{A/\overline{A}} to 𝔐𝔬𝔡_{A}^{∗}. Namely it satisfies the base change property.

A few remarks are in order here. First, in [26], we only considered the case where T = Spec 𝕜. Here we need to study the relative case for the proof of the degeneration formula. When T = Spec 𝕜, we will omit T from the notation. When T is nontrivial, we will call the obstruction theory so defined the relative obstruction theory. Secondly, when we restrict to the subcategory of all triples (B, I, φ) so that B are Artin local rings with residue fields 𝕜, then the above data is the obstruction theory to deforming φ(Spec 𝕜) in S. Thirdly, since Hom_{A}(Ω_{A/\overline{A}} ⊗_{A} B/I, I) is canonically isomorphic to 𝔇𝔢𝔣_{A/\overline{A}}^{1}((B, I)), we have ker\{E^{2∨} → E^{1∨}\} ≅ Ω_{A/\overline{A}}. Lastly, in the definition we can replace E^{•} by its associated complex of sheaves of O_{Spec B}-modules and modify the wording accordingly. This is convenient if we work with obstruction of schemes or stacks. We will call h^{2}(E^{•}) the obstruction module and call its associated sheaf (of O_{Spec B}-modules) obstruction sheaf.

We now investigate the deformation theory of predeformable morphisms to W [n]. Let Γ be the data (g, k, b) representing the genus, the number of marked points and the degree of the maps. Recall that a morphism f:X → W [n] is said to be stable as morphism to 𝔚 if f is an ordinary stable morphism, it is predeformable along the nodal divisors of the fibers of W [n]/A^{n+1} and 𝔄𝔲𝔱(f) is finite. Here 𝔄𝔲𝔱(f) consists of all pairs (a, b) with a: X ≅ X and b ∈ G[n] so that f ∘ a = b ∘ f, where b:W [n] → W [n] is the automorphism defined by the G[n]-action on W [n]. We let 𝔐(W [n], Γ)^{st} be the moduli space of stable morphisms to W [n] of prescribed topological type that are also predeformable as morphisms to the family W [n]/C[n] and are stable as morphisms to the stack 𝔚.

As argued in Section 2 in [23], it is a Deligne-Mumford stack and it comes with a tautological morphism

𝔐(W [n], Γ)^{st} ⟶ 𝔐(𝔚, Γ).

In the remainder of this section, we will cover 𝔐(W [n], Γ)^{st} by affine étale charts and construct canonical obstruction theory of each of these charts. The obstruction theory of 𝔐(𝔚, Γ) will be the descent of the obstruction theory of 𝔐(W [n], Γ)^{st}. Since this study is local, during our study we are free to shrink an open chart S → 𝔐(W [n], Γ)^{st} if necessary.

In the remainder of this subsection we fix an affine étale chart S → 𝔐(W [n], Γ)^{st} with f:X → W [n] its universal family. We let S = Spec A and let D ⊂ X be the divisor of the union of all marked sections of X /S. We fix a collection of charts (U_{α}/V_{α}, f_{α}) of f that covers X satisfying the simplification assumption. In case α is a chart of the second kind, we will reserve the symbols W_{α}, z_{α,i}, w_{α,i}, ϕ_{α}, ψ_{α}, m_{α} and l_{α} for the data associated to the chart α. Our first task is to show that locally there are no obstruction to extending predeformable morphisms f_{α}.

We begin with the notion of flat extensions of an étale neighborhood U_{α}/V_{α}. Let I be an A-module with I_{α} = I ⊗_{A} O_{V_{α}}. We say Ṽ_{α} is an extension of V_{α} by I if V_{α} ⊂ Ṽ_{α} is a subscheme with the ideal sheaf Ĩ of V_{α} ⊂ Ṽ_{α} satisfying (Ĩ)^{2} = 0 and the resulting O_{V_{α}}-modules Ĩ is isomorphic to I_{α}. We say Ũ_{α}/Ṽ_{α} is a small extension of U_{α}/V_{α} by I if Ṽ_{α} is an extension of V_{α} by I and Ũ_{α} is a flat extension of U_{α} over Ṽ_{α}. Now let Ũ_{α}/Ṽ_{α} be an extension of U_{α}/V_{α} by I. We first consider the case where α is a chart of the second kind. By Lemma 1.6, we can assume that the ϕ_{α} of U_{α}/V_{α} in (1.4) is extended to ϕ̃_{α} of Ũ_{α}/Ṽ_{α}. We say an extension f̃_{α}: Ũ_{α} → W [n] of f_{α}: U_{α} → W [n] is a predeformable extension if Ũ_{α} → W [n] → A^{n+1} factor through Ṽ_{α} → A^{n+1} and if the family f̃_{α}:Ũ_{α} → W [n] is predeformable along D_{l_{α}}.

We now define the space Hom_{U_{α}}(f^{∗}Ω_{W [n]}, I)^{†} that will parameterize all such extensions. The group Hom_{U_{α}}(f^{∗}Ω_{W [n]}, I)^{†} is the set of triples

(φ, η_{1}, η_{2}) ∈ Hom_{U_{α}}(f^{∗}Ω_{W [n]}, I_{α}) ⊕ I_{α}^{⊕2}, I_{α} = I ⊗_{A} O_{U_{α}} (1.10)

that obey the following condition:

φ(f^{∗}dw_{α,i}) = f^{∗}(w_{α,i}) · η_{i}, φ(f^{∗}dt_{l}) ∈ I_{α} and η_{1} + η_{2} ∈ I_{α} (1.11)

for i = 1, 2 and l = 1, …, n+1. Here I_{α} = I⊗_{A}O_{V_{α}}. Since f_{α}(U_{α}) ⊂ W_{α} and w_{α,i} ∈ Γ(O_{W_{α}}), f^{∗}dw_{α,i} ∈ f^{∗}Ω_{W [n]} ⊗_{O_{X}} O_{U_{α}} and hence φ(f^{∗}dw_{α,i}) makes sense. Note also that because t_{l_{α}} = w_{α,1}w_{α,2}, from (1.11) we have

s_{α}^{m_{α}}(η_{1} + η_{2}) = φ(f^{∗}dt_{l_{α}}). (1.12)

Clearly, Hom_{U_{α}}(f^{∗}Ω_{W [n]}, I)^{†} is an A-module and is A-flat if I is A-flat.

When α is a chart of the first kind, we define Hom_{U_{α}}(f^{∗}Ω_{W [n]}, I)^{†} be the subgroup of φ ∈ Hom_{U_{α}}(f^{∗}Ω_{W [n]}, I_{α}) such that φ(dt_{l}) ∈ I_{α} for all l. Note that in case α and β are two charts of f and p_{α} : U_{αβ} ≜ U_{α} ×_{X} U_{β} → U_{α} is the projection, then there is a canonical restriction A-homomorphism

p_{α}^{∗} : Hom_{U_{α}}(f^{∗}Ω_{W [n]}, I)^{†} ⟶ Hom_{U_{αβ}}(f^{∗}Ω_{W [n]}, I)^{†}.

We now state and prove the following local deformation lemma.

Lemma 1.12. Let Ũ_{α}/Ṽ_{α} be an extension of U_{α}/V_{α} by I. Then f_{α} automatically extends to a predeformable morphism f̃_{α}:Ũ_{α} → W [n]. Further, after fixing one such extension, say f̃′_{α}, the space of all such extensions is canonically isomorphic to the space Hom_{U_{α}}(f^{∗}Ω_{W [n]}, I)^{†}.

Proof. We will consider the case where α is a chart of the second kind. The other case is simpler. First, we extend ϕ_{α} to a parameterization ϕ̃_{α} of Ũ_{α}/Ṽ_{α} with z̃_{α,i} and s̃_{α} the corresponding extensions of z_{α,i} and s_{α}, respectively. We consider the composite ψ_{α} ∘ f_{α}:U_{α} → Θ_{l_{α}}, where Θ_{l_{α}} is defined in (1.1). Let h_{α,i} ≡ 1 ∈ Γ(O_{U_{α}}) and g_{α,l} ∈ Γ(O_{V_{α}}) be part of the definition of the chart of f_{α}, as in (1.5) and (1.6). Since both U_{α} and V_{α} are affine, we can extend h_{α,i} and g_{α,l} to h̃_{α,i} ∈ Γ(OŨ_{α}) and g̃_{α,l} ∈ Γ(OṼ_{α}), respectively, so that h̃_{α,1}h̃_{α,2} ∈ Γ(OṼ_{α}) and g̃_{α,l_{α}} = s_{α}^{m_{α}}(h̃_{α,1}h̃_{α,2}). We then define F : Ũ_{α} → Θ_{l_{α}} by w_{α,i} ↦ z_{α,i}^{m_{α}} h̃_{α,i} and t_{l} ↦ g̃_{α,l}. Since W_{α} → Θ_{l_{α}} is smooth, we can lift F to an extension f̃_{α} : Ũ_{α} → W [n] of f_{α}. Namely, f̃_{α} is an extension of f_{α} such that ψ_{α} ∘ f̃_{α} = F. The morphism f̃_{α} is a desired extension.

Now let f̃′_{α} be a fixed predeformable extension of f_{α} to Ũ_{α}/Ṽ_{α}. Let f̃_{α} be any predeformable extension of f_{α}. As morphisms from Ũ_{α} to Θ_{l_{α}}, ψ_{α} ∘ f̃_{α} is defined by

(ψ_{α} ∘ f̃_{α})^{∗}(w_{α,i}) = z_{α,i}^{m_{α}} h̃_{α,i} and (ψ_{α} ∘ f̃_{α})^{∗}(t_{l}) = g̃_{α,l} (1.13)

and ψ_{α} ∘ f̃′_{α} is defined by

(ψ_{α} ∘ f̃′_{α})^{∗}(w_{α,i}) = z_{α,i}^{m_{α}} h̃′_{α,i} and (ψ_{α} ∘ f̃′_{α})^{∗}(t_{l}) = g′_{α,l}. (1.14)

We let η_{i} = h̃_{α,i} − h̃′_{α,i} ∈ I_{α}. Then η_{1} + η_{2} ∈ I_{α} since h̃_{α,1}h̃_{α,2} and h̃′_{α,1}h̃′_{α,2} ∈ Γ(OṼ_{α}). Now let φ ∈ Hom_{U_{α}}(f_{α}^{∗}Ω_{W}, I) be defined by the difference^{6} d(f̃_{α} − f̃′_{α}). It can be easily checked, based on (1.13) and (1.14), that (φ, η_{1}, η_{2}) is in Hom_{U_{α}}(f^{∗}Ω_{W [n]}, I)^{†}.

It remains to check that this correspondence is one-one and onto, which is straightforward. This completes the proof of the lemma. q.e.d.

Remark 1.13. In the remainder part of this paper, we will call (φ, η_{1}, η_{2}) the log difference of f̃_{α} and f̃′_{α}, denoted by d^{†}(f̃_{α} − f̃′_{α}). Furthermore, if f̃_{1}, f̃_{2} and f̃_{3} are three predeformable extensions of f_{α} to Ũ_{α}, then d^{†}(f̃_{3} − f̃_{1}) = d^{†}(f̃_{3} − f̃_{2}) + d^{†}(f̃_{2} − f̃_{1}).

6Let ι : A′ → A be a small ring extension with I = ker{ι}. We call A′ → A a small extension if I^{2} = 0. Let B be any ring. Let f_{1}, f_{2} : B → A be two ring homomorphisms so that ι ∘ f_{1} = ι ∘ f_{2}. We define the difference of f_{1} and f_{2} to be the map d(f_{1} − f_{2}) : Ω_{B} → I defined by b ⊗ 1 − 1 ⊗ b ↦ f_{1}(b) − f_{2}(b) ∈ I. It is an A-homomorphism Ω_{B} ⊗_{B} A → I. Note that once f_{1} is fixed, then f_{2} is uniquely determined by d(f_{1} − f_{2}). Conversely, any φ ∈ Hom_{A}(Ω_{B} ⊗_{B} A, I) defines a unique homomorphism f_{2} :B → A so that d(f_{1} − f_{2}) = φ.

We next study the obstruction to deforming a global predeformable morphism f:X → W [n]. Before we proceed, we introduce more convention which will be followed throughout this paper. We let ι_{α}: U_{α} → X and ȷ_{α} : V_{α} → S be the tautological projections. As a convention, for each α ∈ Λ we let A_{α} = Γ(V_{α}) with A → A_{α} the tautological homomorphism, for α and β ∈ Λ we will denote by U_{αβ} the product U_{α} ×_{X} U_{β} and by p_{α} the projection U_{αβ} → U_{α}. The same convention applies to multi-indices in the obvious way. We let π_{α}:U_{α} → V_{α} be the projection. For sheaf of O_{X} -modules A (resp. A-module I; divisor D) we will denote by A_{α} (resp. I_{α}, resp. D_{α}) the pullback sheaf ι_{α}^{∗}A (resp. I ⊗_{A} A_{α}; resp. D ×_{X} U_{α}). Also, for A-module I, we will use I (resp. I_{α}) to denote the sheaf of O_{X} -modules O_{X} ⊗_{A} I (resp. O_{U_{α}} ×_{A} I). We denote by d_{α} : F_{α}^{0} → F_{α}^{2} the pullback of d: F^{0} → F^{1} in (1.15). For sheaves of O_{X} -modules A and B we agree that Hom_{U_{α}}(A, B) = Hom_{U_{α}}(A_{α}, B_{α}) and for A-modules I and J we agree Hom_{A_{α}}(I, J) = Hom_{A_{α}}(I_{α}, J_{α}).

We now study the deformation of f:X /S → W [n]/A^{n+1}. Since X /S is a flat family of nodal curves, there is a complex of free A-modules

F^{•} = [F^{0} ⟶^{d} F^{1}] (1.15)

so that for any A-module I,

Ext_{X}^{i} (Ω_{X /S}(D), I) = h^{i}(F^{•} ⊗_{A} I). (1.16)

Now let S̃ ≜ Spec A ∗ F^{1∨} ^{7} with S ⊂ S̃ be the immersion induced by the obvious projection A ∗ F^{1∨} → A. Let 1 ∈ F^{1} ⊗_{A} F^{1∨} be the identity element and let [1] ∈ Ext_{X}^{1} (Ω_{X /S}(D), F^{1∨}) be the associated element. The element [1] defines a family X̃/S̃ extending the family X /S, using (1.16). It has the following properties:

Firstly, let I be any A-module and let a ∈ F^{1} ⊗_{A} I be any element. Let T ≜ Spec A ∗ I and X_{T} /T be the extension of X defined by the cohomology class [a] ∈ Ext_{X}^{1} (Ω_{X /S}(D), I) of a ∈ F^{1} ⊗_{A} I. Then X̃ ×S̃ T ≅ X_{T} under the morphism T → S̃ defined by A ∗ F^{1∨} → A ∗ I via (x, y) ↦ (x, a(y)).

Secondly, let T̃ ≜ Spec A∗F^{0∨} → S̃ = Spec A∗F^{1∨} be the morphism defined by A ∗ F^{1∨} → A ∗ F^{0∨} via (x, y) ↦ (x, d^{∨}(y)). Then we have isomorphism over T̃ X̃ ×S̃ T̃ ≅ X ×_{S} T̃, where the projection T̃ → S is defined by the obvious inclusion A ↪ A ∗ F^{0∨}.

7In this section we will denote by A ∗ I the trivial ring extension of A by the A-module I.

As mentioned, 1 ∈ F^{1} ⊗_{A} F^{1∨} defines an extension X̃/S̃, which we fix from now on. We let Ũ_{α} (resp. Ṽ_{α}) be the étale neighborhood of X̃ (resp. S̃) that is the minimal extension of U_{α} (resp. V_{α})^{8} . Then {Ũ_{α}/Ṽ_{α}} forms a covering of X̃/S̃. By Lemma 1.12, f_{α} : U_{α} → W_{α} can be extended to a predeformable morphism Ũ_{α} → W [n]. For each α we pick one such extension ζ_{α} : Ũ_{α} → W [n] once and for all. Now let α, β ∈ Λ be a pair so that U_{αβ} ≠ ∅. Let Ũ_{αβ} = Ũ_{α} ×X̃ Ũ_{β} and let p̃_{α}: Ũ_{αβ} → Ũ_{α} be the projection, following our convention. Since both ζ_{α} ∘ p̃_{α} and ζ_{β} ∘ p̃_{β} are predeformable extensions of f_{αβ} : U_{αβ} → W [n], by Lemma 1.12 their difference defines an element

ζ_{αβ} ≜ d^{†}(ζ_{β} ∘ p̃_{β} − ζ_{α} ∘ p̃_{α}) ∈ Hom_{U_{αβ}}(f^{∗}Ω_{W [n]}, F^{1∨})^{†}.

This defines a homomorphism

ζ_{αβ}(·) : F^{1} ⟶ Hom_{U_{αβ}}(f^{∗}Ω_{W [n]}, A)^{†}. (1.17)

Now to each α we construct a homomorphism

ζ_{α}(·) : F^{0} ⟶ Hom_{U_{α}}(f^{∗}Ω_{W [n]}, A)^{†}. (1.18)

Let 1 ∈ F^{0} ⊗_{A} F^{0∨} be the identity element. Then

d(1) ∈ F^{1} ⊗_{A} F^{0∨} ≡ Hom_{A}(F^{1∨}, F^{0∨})

(d is the differential in the complex F^{•}) defines a morphism

τ : T ≜ Spec A ∗ F^{0∨} → Spec A ∗ F^{1∨} = S̃.

Let X_{T} /T be the pullback family of X̃/S̃ via τ with q : X_{T} → X̃ the induced the projection. By the second property after (1.16), X_{T} is isomorphic to X ×_{S} T for the T → S induced by the obvious inclusion A → A ∗ F^{0∨}. We let q_{0} : X_{T} → X be the induced projection. Clearly, U_{α} ≜ Ũ_{α} ×S̃ T is canonically isomorphic to U_{α} ×_{S} T under the isomorphism X_{T} ≅ X ×_{S} T. Then restricting to U_{α}, both ζ_{α} ∘ q and f ∘ q_{0} are predeformable extensions of f_{α}. By Lemma 1.12, the difference

d^{†}(ζ_{α} ∘ q − f ∘ q_{0}) ∈ Hom_{U_{α}}(f^{∗}Ω_{W [n]}, F^{0∨})^{†}.

It defines the desired homomorphism ζ_{α}(·) in (1.18).

8We say Ũ_{α} is a minimal extension of the étale neighborhood U_{α} if Ũ_{α} → X̃ is an étale neighborhood and Ũ_{α} ×X̃ X ≅ U_{α}.

Lemma 1.14. Both ζ_{α}(·) and ζ_{αβ}(·) are homomorphisms of Amodules. Further, for any a ∈ F^{0} we have

−p_{α}^{∗}ζ_{α}(a) + p_{β}^{∗}ζ_{β}(a) = ζ_{αβ}(d(a)) ∈ Hom_{U_{αβ}}(f^{∗}Ω_{W [n]}, A)^{†},

where p_{α}:U_{αβ} → U_{α} is the projection and p_{α}^{∗}ζ_{α}(·) is the pullback homomorphism.

Proof. The proof is straightforward and will be omitted. q.e.d.

In the next part, we will construct the complex that will be a part of the perfect obstruction theory of S ⊂ 𝔐(W [n], Γ)^{st} we are about to construct. Let I be any A-module. We let D(I)^{•} be the Čech complex

D(I)^{•} = C^{•}(Λ, Hom(f^{∗}Ω_{W [n]}, I)^{†})

associating to the covering Λ, where

Γ(U_{α_{1}...α_{k}}, Hom(f^{∗}Ω_{W [n]}, I)^{†}) = Hom_{U_{α_{1}...α_{k}}} (f^{∗}Ω_{W [n]}, I)^{†}

with ∂:D(I)^{•} → D(I)^{•+1} the coboundary differential in the Čech complex. We let

δ_{k} : F^{k} ⊗_{A} I ⟶ C^{k}(Λ, Hom(f^{∗}Ω_{W [n]}, I)^{†}), k = 0, 1

be defined by δ_{0}(a)_{α} = ζ_{α}(a) and δ_{1}(b)_{αβ} = ζ_{αβ}(b). By Lemma 1.14, δ_{k} are homomorphisms of A-modules. We now show that this defines a homomorphism of complexes

δ : F^{•} ⊗_{A} I ⟶ D(I)^{•}.

To prove this, we need to check that δ_{1} ∘ d = ∂ ∘ δ_{0} on F^{0} ⊗_{A} I and ∂ ∘ δ_{1} = 0 on F^{1} ⊗_{A} I. We will check ∂ ∘ δ = 0 on F^{1} ⊗_{A} I and leave the other to the readers. Let b ∈ F^{1} ⊗_{A} I be any element. By definition,

(∂ ∘ δ_{1})(b)_{αβγ} = δ_{1}(b)_{αβ} − δ_{1}(b)_{αγ} + δ_{1}(b)_{βγ} = ζ_{αβ}(b) − ζ_{αγ}(b) + ζ_{βγ}(b)

as elements in Hom_{U_{αβγ}}(f^{∗}Ω_{W [n]}, I)^{†}, where the summation is taken after pulling back each term in the summation to this module in the obvious way. But this vanishes because of the relation ζ_{βγ} −ζ_{αγ} +ζ_{αβ} = 0, following Remark 1.13. This shows that ∂ ∘ δ_{1} = 0 on F^{1} ⊗_{A} I.

In the end, we define the complex E(I)^{•} by

E(I)^{k} = ⊕_{i+j=k}(F^{i} ⊗_{A} I ⊕ D(I)^{j−1}) (1.19)

with the differential d_{E} :E(I)^{k} → E(I)^{k+1} defined via

F^{0} ⊗_{A} I ⟶^{(d⊕,−δ_{0})} F^{1} ⊗_{A} I ⊕ D(I)^{0} ⟶^{δ_{1}+∂} D(I)^{1} ⟶^{∂} …

When I = A, we will abbreviate D(A)^{•} and E(A)^{•} to D^{•} and E^{•} respectively.

Lemma 1.15. We assume Λ is a sufficiently fine cover of f. Then for any A-module I, E(I)^{•} is canonically isomorphic to E^{•} ⊗_{A} I as complexes. Further, the complex E^{•} is a complex of flat A-modules.

Proof. The proof is straightforward and will be omitted. q.e.d.

We remark that the complex E^{•} just constructed depends on the choice of the atlas Λ. To emphasize this dependence we shall denote it by E_{Λ}^{•}. Let Λ′ be any atlas of f that is a refinement of Λ with associated complex E_{Λ′}^{•}. Following [30, III.2], we can define a homomorphism of complexes E_{Λ}^{•} → E_{Λ′}^{•} which induces a map of cohomologies

ρ(Λ, Λ′) : h^{i}(E_{Λ}^{•} ⊗_{A} I) ⟶ h^{i}(E_{Λ′}^{•} ⊗_{A} I)

and then form the direct limit lim_{→} h^{i}(E_{Λ} ⊗_{A} I) taken over all charts of f. Note that this limit is a functor from 𝔐𝔬𝔡_{A} to 𝔐𝔬𝔡_{A}, denoted by 𝔥^{i}(E^{•}). We now assume Λ is fine enough so that 𝔥^{i}(E^{•})(I) = h^{i}(E_{Λ}^{•}⊗_{A}I) for all A-module I. We fix such a Λ once and for all, and abbreviate the resulting complex E_{Λ}^{•} to E^{•}.

We now prove the main result of this section.

Proposition 1.16. Let 𝔇𝔢𝔣_{A}^{1} be the functor of the first order deformations of morphisms to S = Spec A, which is naturally a functor from A-modules to A-modules. Then 𝔇𝔢𝔣_{A}^{1} is naturally isomorphic to the functor 𝔥^{1}(E^{•}).

Proof. Let ξ = (B, I) be any object in 𝔐𝔬𝔡_{A}. We first show that there is a canonical isomorphism 𝔇𝔢𝔣_{A}^{1}(ξ) ≅ h^{1}(E^{•} ⊗_{A} I).

Let T = Spec B and T̃ = Spec B ∗ I, the trivial extension by I. Since B is an A-algebra, there is a tautological morphism T → S. Let x ∈ 𝔇𝔢𝔣_{S}^{1}(ξ) (= 𝔇𝔢𝔣_{A}^{1}(ξ)) be any element, associated to an extension T̃ → S of T → S, and let f_{T} : X_{T} → W [n] (resp. fT̃ : XT̃ → W [n]) be the pull back family of f via T → S (resp T̃ → S). We let D_{T} ⊂ X_{T} and DT̃ ⊂ XT̃ be the associated divisors of marked points and let I = O_{X_{T}} ⊗_{B} I. First of all, since XT̃ is a flat extension of X_{T} to T̃, it associates to a unique element

[a] ∈ Ext_{X_{T}}^{1} (Ω_{X_{T} /T} (D_{T}), I) ≅ h^{1}(F^{•} ⊗_{A} I), (1.20) where a ∈ F^{1} ⊗_{A} I. Now let S̃ be the trivial extension of S by F^{1∨} and let X̃/S̃ be the family defined after (1.16). Then a defines a morphism φ_{a} : T̃ → S̃ via the homomorphism A ⊕ F^{1∨} → B ⊕ I defined by (x, y) ↦ (\overline{x}, a(y)), where \overline{x} is the image of x in B. By the first property after (1.16), a defines an isomorphism γ_{a} : XT̃ ≅ X̃ ×S̃ T̃, where the projection T̃ → S̃ is via φ_{a}. We let Ũ_{α} = Ũ_{α} ×S̃ T̃ and Ṽ_{α} = Ṽ_{α} ×S̃ T̃. Then each Ũ_{α}/Ṽ_{α} extends to a chart (Ũ_{α}/Ṽ_{α}, f_{T̃,α}) of fT̃, where f_{T̃,α} ≜ fT̃|Ũ. We let ρ̃_{α} : Ũ_{α} → Ũ_{α} be the projection. Then over each Ũ_{α} we have two predeformable extensions of f_{T}: One is f_{T̃,α} and the other is the composite ζ_{α} ∘ ρ̃_{α}. Let

b_{α} = d^{†}(f_{T̃,α} − ζ_{α} ∘ ρ̃_{α}) ∈ Hom_{U_{α}}(f_{T}^{∗}Ω_{W [n]}, I)^{†}

and let b = {b_{α}} ∈ D^{0} ⊗_{A} I. We claim that (a, b) ∈ E^{1} ⊗_{A} I is in the kernel of d_{E}. For this, we only need to check that

d_{E}(a, b)_{αβ} = δ(a)_{αβ} + ∂(b)_{αβ}

= ζ_{αβ}(a) + (b_{β} − b_{α}) = ζ_{αβ}(a) − d^{†}(ζ_{β} ∘ ρ̃_{β} − ζ_{α} ∘ ρ̃_{α})

vanishes for all pairs (α, β). But this follows immediately from the definition of ζ_{αβ}(·). This shows that (a, b) defines a cohomology class [(a, b)] ∈ h^{1}(E^{•} ⊗_{A} I).

Next, we show that [(a, b)] is independent of the choices of a and the isomorphisms γ_{a}: XT̃ ≅ X̃ ×S̃ T̃. Let a′ ∈ F^{1} ×_{A} I be another element so that [a′] = [a] in (1.20). Then a − a′ = d_{I}(c) for a c ∈ F^{0} ⊗_{A} I, where d_{I}:F^{0} ⊗_{A} I → F^{1} ⊗_{A} I is induced by d in (1.15). Now let

φ_{a′} : T̃ → S̃, γ_{a′} : XT̃ ≅ X̃ ×S̃ T̃,
ρ′_{α} : Ũ_{α} → Ũ_{α}, b′_{α} = d^{†}(f_{T̃,α} − ζ_{α} ∘ ρ′_{α})

be objects defined similarly with a replaced by a′. Let b′ = {b′_{α}}. We claim that

d_{E}(c) = (a, b) − (a′, b′) ∈ E^{1} ⊗_{A} I.

Once this is established then [(a, b)] = [(a′, b′)] ∈ h^{1}(E^{•} ⊗_{A} I), which shows that [(a, b)] only depends on the class x ∈ 𝔇𝔢𝔣_{S}^{1}(ξ). This way we obtain a map

T(ξ) : 𝔇𝔢𝔣_{S}^{1}(v) ⟶ h^{1}(E^{•} ⊗_{A} I).

Now we prove the claim. Since

d_{E}(c) = (∂(c), −δ(c)) = (d_{I}(c), −δ(c))

and d_{I}(c) = a − a′, it suffices to show that δ(c)_{α} = ζ(c)_{α} is identical to

−(b − b′)_{α} = −(b_{α} − b′_{α}) = −d^{†}(f_{T̃,α} − ζ_{α} ∘ ρ̃_{α}) + d^{†}(f_{T̃,α} − ζ_{α} ∘ ρ′_{α})
= d^{†}(ζ_{α} ∘ ρ̃_{α} − ζ_{α} ∘ ρ′_{α}).

The proof that ζ_{α}(c) = d^{†}(ζ_{α} ∘ ρ̃_{α} − ζ_{α} ∘ ρ′_{α}) is routine and will be omitted.

We now show that the map T(ξ) is one-one and onto. We give an outline of the proof since it is standard. We first show that it is one-one. Let x ∈ 𝔇𝔢𝔣_{S}^{1}(ξ) be any element so that T(ξ)(x) = 0 ∈ h^{1}(E^{•} ⊗_{A} I). We let (a, b) be the pair constructed associated to the family fT̃:XT̃ → W [n] following the previous discussion. Since [(a, b)] = 0, there is a c ∈ F^{0} ⊗_{A} I so that (a, b) = d_{E}(c). This implies at first that a = d_{I}(c). Hence XT̃ ≅ X_{T} ×_{T} T̃ under the obvious projection T̃ → T via B → B ⊕ I. Let ρ : XT̃ → X_{T} be the projection. Because T(ξ)(x) is well-defined, T(ξ)(x) is also represented by (0, b′), where b′ = {b′_{α}} and

b′_{α} = d^{†}(f_{T̃,α} − f_{T,α} ∘ ρ_{α}) ∈ Hom_{U_{α}}(f_{T}^{∗}Ω_{W [n]}, I)^{†}.

Again, since T(ξ)(x) = 0, there must be a c ∈ F^{0} ⊗_{A} I so that d_{I}(c) = 0 and δ(c) = b′. Hence c lifts to an element in Ext_{X_{T}}^{0} (Ω_{X_{T} /T} (D_{T}), I), which defines a new isomorphism γ:XT̃ ≅ X_{T} ×_{T} T̃. Further, δ(c) = b′ implies that under this new isomorphism fT̃ is the constant extension of f_{T}. Namely, fT̃ is the pull back of f_{T} via the projection T̃ → T. This proves that x = 0 in 𝔇𝔢𝔣_{S}^{1}(ξ) and hence T(ξ) is one-one.

The onto part is similar. Since this argument is standard in deformation theory, we will omit it here. In the end, we need to check that T(ξ) is a homomorphism of modules and that T is an isomorphism of functors. This is straightforward and will be omitted. q.e.d.

Corollary 1.17. Let ξ = (B, I, φ_{0}) be any object in 𝔗𝔯𝔦_{S}. Suppose 𝔇𝔢𝔣_{S}(ξ) ≠ ∅, then 𝔇𝔢𝔣_{S}(ξ) is isomorphic to the set h^{1}(E^{•} ⊗_{A} I).

Proof. Since S is an étale chart of 𝔐(W [n], Γ)^{st}, which is a Deligne-Mumford stack, the standard fact in deformation theory shows that once an extension φ: Spec B → S of φ_{0} is fixed, then the space of all such extensions is canonically isomorphic to 𝔇𝔢𝔣_{S}^{1}(ξ_{0}) ≅ h^{1}(E^{•} ⊗_{A} I), where ξ_{0} = (B/I ∗ I, I, φ_{0}). This proves the corollary. q.e.d.

Proposition 1.18. There is a natural obstruction theory to deformation of the family of predeformable f : X → W [n] over S that takes values in 𝔥^{2}(E^{•}).

Proof. The construction of the obstruction class is standard, as shown in [25] for smooth targets. Let ξ = (B, I, φ) be an object in 𝔗𝔯𝔦_{S}. Let T = Spec B/I and let f_{T} : X_{T} → W [n] be the pullback of f via T → S. By definition, extending T → S to T̃ → S is equivalent to extending f_{T} to a family of predeformable morphisms over T̃. We now construct the obstruction class to this extension problem.

Since deformation of pointed curves is unobstructed, we can extend X_{T} to a family over T̃. By our choice of F^{•}, such extension can be realized by an extension T̃ → S̃ of T → S. We let XT̃ be the pullback family X̃ ×S̃ T̃. Let Ũ_{α} = Ũ_{α} ×S̃ T̃, Ṽ_{α} = Ṽ_{α} ×S̃ T̃, U_{α} = Ũ_{α} ×_{XT̃} X_{T} and let V_{α} = Ṽ_{α}×_{XT̃} X_{T}. Also we extend U_{α}/V_{α} to a chart (U_{α}/V_{α}, f_{T,α}, W_{α}) of f_{T}. This way Ũ_{α}/Ṽ_{α} is a minimal extension of U_{α}/V_{α} to XT̃/T̃. Next we let ρ̃_{α}: Ũ_{α} → Ũ_{α} be the projection. For any α we pick a family of predeformable extension h̃_{α}: Ũ_{α} → W [n] of f_{T,α}. We let p̃_{α}: Ũ_{αβ} → Ũ_{α} be the projection. We then let

b_{αβ} = −d^{†}(h̃_{β} ∘ p̃_{β} − h̃_{α} ∘ p̃_{α}) ∈ Hom_{U_{αβ}}(f_{T}^{∗}Ω_{W [n]}, I)^{†}. (1.21)

We let b = {b_{αβ}}, which belongs to D^{1} ⊗_{A} I ⊂ E^{2} ⊗_{A} I. It follows from the Remark 1.13 that b is a cocycle, and thus defines a cohomology class [b] in h^{2}(E^{•} ⊗_{A} I). The technical part of the proof is to check that the cohomology class [b] is independent of the choices of T̃ → S̃ and h̃_{α}. The argument for this is straightforward though tedious, and will be omitted.

We now show that it is an obstruction class to extending f_{T} to families of predeformable morphisms over T̃. First, if such extensions do exist, then we can choose XT̃ and h̃_{α} be data coming from one of such extensions. Then the corresponding b′ = 0 as cocycle and thus [b] = [b′] = 0. This shows that [b] = 0 whenever extensions of f_{T} exist. Now assume [b] = 0. We first look at the case where the cycle b ∈ D^{1} ⊗_{A} I is a coboundary in D^{•} ⊗_{A} I. Namely, there is a c ∈ D^{0} ⊗_{A} I so that b = ∂(c). Let c = {c_{α}} with c_{α} ∈ Hom_{U_{α}}(f_{T}^{∗}Ω_{W [n]}_{†}, I)^{†}. Then

by Lemma 1.12, we can find predeformable extension f_{T̃,α}:Ũ_{α} → W [n] of f_{T,α} so that d^{†}(f_{T̃,α} − h̃_{α}) = c_{α}. Then over Ũ_{αβ}, the difference of the pullbacks f_{T̃,β} ∘ p̃_{β} and f_{T̃,α} ∘ p̃_{α} is

d^{†}(f_{T̃,β} ∘ p̃_{β} − f_{T̃,α} ∘ p̃_{α})
= d^{†}(f_{T̃,β} ∘ p̃_{β} − h̃_{β} ∘ p̃_{β}) − d^{†}(f_{T̃,α} ∘ p̃_{α} − h̃_{α} ∘ p̃_{α})
+ d^{†}(h̃_{β} ∘ p̃_{β} − h̃_{α} ∘ p̃_{α})
= (c_{β} − c_{α}) − b_{αβ} = 0.

Hence {f_{T̃,α}} patch together to form a desired extension XT̃ → W [n] of f_{T}.

In general, assume [b] = 0, then there is an a ∈ F^{0} ⊗_{A} I and c ∈ D^{0} ⊗_{A} I so that

b = d_{E}(a, c) = δ(a) + ∂(c).

Now let r′: T̃ → S̃ be a new extension of T → S so that d(r′ − r̃) = a, where r̃ : T̃ → S̃ is the morphism used to construct the cycle b. One checks that if one uses the new extension X̃ ×S̃ T̃, where T̃ → S̃ is via r′, to construct a similar cycle b′ ∈ D^{1} ⊗_{A} I, then b′ is a coboundary in D^{•} ⊗_{A} I. This reduces the situation to the previous case studied, and hence confirms that a predeformable extension of f_{T} over T̃ can be found. This shows that [b] is an obstruction class to extending f_{T} to families of predeformable morphisms over T̃, or equivalently the obstruction class to extending T → S to T̃ → S.

We define

𝔬𝔟(B, I, φ) = [b] ∈ h^{2}(E^{•} ⊗_{A} I).

Again it is direct to check that this assignment defines a functor from 𝔗𝔯𝔦_{S} to 𝔐𝔬𝔡_{S}^{∗}. This completes the proof of Proposition 1.18. q.e.d.

We now summarize the results of this section in the following theorem. We need a vanishing lemma whose proof will be provided in Proposition 5.1.

Lemma 1.19. For sufficiently fine Λ, we have h^{i}(D^{•} ⊗_{A} I) = 0 for any A-module I and i ≥ 2.

Theorem 1.20. Let S be an affine chart of the moduli stack 𝔐(W [n], Γ)^{st}. Then the obstruction theory just defined is a perfect obstruction theory of S.

Proof. Let f:X → W [n] be the universal family over S. If suffices to show that there is a complex of finitely generated free A-modules E^{•} = [E^{1} → E^{2}] so that it is quasi-isomorphic to E^{•}, where E^{•} is the complex associated to a sufficiently find atlas Λ of f. Since h^{i}(D^{•} ⊗_{A} I) = 0 for i ≥ 2 and any I, h^{i}(E^{•} ⊗_{A} I) = 0 for i ≥ 3 and any I. Hence there is a bounded subcomplex Ẽ^{•} of flat A-modules so that it is quasi-isomorphic to E^{•}. Then we can apply the standard technique [12, III.12] to find a bounded subcomplex E^{•} of finitely generated free A-modules that is quasi-isomorphic to E^{•}. Finally, since h^{i}(E ⊗_{A} I) ≠ 0 only for i = 1 and 2, we can choose E^{•} to be of the form [E^{1} → E^{2}]. This proves the theorem. q.e.d.

Remark 1.21. In the Appendix, we will express the cohomology h^{•}(E^{•}) in terms of some known cohomologies.

1.3 Obstruction to deforming relative stable morphisms

We will follow the notation developed in [23, Section 4] concerning relative stable morphisms. Let (Z, D) be a polarized relative pair and let Z[n]^{rel} be the expanded relative pair constructed there. Recall that Z[n]^{rel} consists of a proper variety Z[n] over A^{n} and a smooth divisor D[n] ⊂ Z[n] that is isomorphic to D × A^{n} under the projection Z[n] → Z × A^{n}. The pair (Z[n], D[n]) over A^{n} also admits an equivariant G[n] action whose action on A^{n} is the standard one^{9} . The fibers of Z[n]/A^{n} has at most normal crossing singularities and the singular locus of all the fibers of Z[n]/A^{n} is a disjoint union of smooth varieties B_{1}, …, B_{n}, indexed so that B_{l} surjects to the l-th coordinate hyperplane H_{l} ⊂ A^{n}.

In [23] we used admissible graph to describe the topological type of relative stable morphisms to (Z[n], D[n]). Recall that a weighted graph Γ consists of a collection of vertices V_{Γ}, an ordered collection of weighted roots R_{Γ} and an ordered collection of legs L_{Γ} plus two weight functions g, b: V_{Γ} → ℤ_{≥0} and a multiplicity assignment μ: R_{Γ} → ℤ^{+}. We require Γ to be relatively connected in the sense that either Γ is connected or each of its vertex has at least one root attached to it.

A relative morphism to Z[n]^{rel} of type Γ consists of a pointed complete nodal curve X and a morphism f:X → Z[n] that has the following property: First the marked points of X are labeled by the ordered legs and roots of Γ, say are p_{1}, …, p_{k} ∈ X and q_{1}, …, q_{r} ∈ X. We let μ_{1}, …, μ_{r} be the the weights of the respective roots of Γ. Secondly, the connected components of X are labeled by a ∈ V_{Γ} and the arithmetic genus of the component X_{a} is g(a). Further, in case a root or a leg is attached to a vertex a then its associated marked point must lie in the connected component X_{a}. Thirdly, restricting to each connected component X_{a} the morphism f|_{X_{a}} with all the marked points in X_{a} is an ordinary stable morphism to Z[n] of degree b(a) (using the polarization on Z chosen implicitly). Lastly, as divisor f^{−1}(D[n]) = μ_{1}q_{1}+⋯+μ_{r}q_{r}.

We recall the notion of relative stable morphisms to ℨ^{rel} defined in [23]. Let f : X → Z[n] be a relative morphism of type Γ, as described. Recall fibers of Z[n]/A^{n} have normal crossing singularities along B_{1}, …, B_{n}. We say f is predeformable if it is predeformable along all B_{i}, as defined before. We say f is stable as a relative morphism to ℨ^{rel} if f is predeformable and 𝔄𝔲𝔱(f) is finite. Here 𝔄𝔲𝔱(f) is the group of all pairs (a, b) where a: X → X are automorphisms and b ∈ G[n] so that f ∘ a = f^{b}. (Here we view b as an automorphism b:Z[n] → Z[n] using the G[n] action on Z[n] and f^{b} is the composite of f with b.) As was proved in [23], the moduli of all relative morphisms to Z[n]^{rel} of type Γ that are stable as morphisms to ℨ^{rel} form a Deligne-Mumford stack. We will denote this stack by 𝔐(Z[n]^{rel}, Γ)^{st}. The goal of this subsection is to describe the obstruction theory of this moduli stack.

9In short, Z[1] is the blowing up of Z × A^{1} along D × 0 and D[1] is the proper transform of D × A^{1}. Z[2] is the blowing up of Z[1] × A^{1} along D[1] × A^{1}, etc. The G[n] action is the unique lifting of its standard action on Z × A^{n}.

Similar to the case f : X → W [n], for any family of relative stable morphisms f : X → Z[n] over S there is a canonical log structure on X /S and on Z[n]/A^{n} that makes f a morphism between log schemes. The log structure on A^{n} (resp. Z[n]) is given by the divisor ∪_{l=1}^{n}H_{l} ⊂ A^{n} (resp. Z[n] ×_{A^{1}} 0 ∪ D[n] ⊂ Z[n]). As to the log structure on X /S, we first note that if we let W/A^{1} be Z[1]/A^{1}, then Z[n]/A^{n} = W [n−1]/A^{n} and f is a family of predeformable morphisms to W [n − 1]. We let (X , M′) and (S, N) be the associated log structures of f:X → W [n−1]. We let M′′ → O_{X} be the log structure of f^{−1}(D[n]) ⊂ X. The the desired log structure on X is the associated log structure of the prelog structure M′⊕M′′ → O_{X}. It is obvious that this gives X a log structure M, making it a log scheme over (S, N) and making f a log morphism between X^{†}/S^{†} → Z[n]^{†}/A^{n†}.

Let S = Spec A be an affine chart of 𝔐(Z[n]^{rel}, Γ)^{st} with f : X → Z[n] the universal family and q_{i}, p_{j} : S → X its marked sections. We cover f by charts of the first or the second kind. Let these charts be (U_{α}/V_{α}, f_{α}, Z_{α}) indexed by Λ as defined before (1.5) with W [n] (resp. D_{l}; resp. n + 1) replaced by Z[n] (resp. B_{l}; resp. n). We let D ⊂ X be the divisor of the locus of all marked sections of X /S. We now fix a complex of finite rank free A-modules F^{•} = [F^{0} ⟶^{d} F^{1}] so that

h^{•}(F^{•} ⊗_{A} I) = Ext_{X}^{•} (Ω_{X /S}(D), I), I = I ⊗_{O_{S}} O_{X}

for all A-module I. We form the group Hom_{U_{α}}(f^{∗}Ω_{Z[n]}, I)^{†} as follows: In case U_{α}/V_{α} is a chart away from f^{−1}(D[n]), this group is Hom_{U_{α}}(f^{∗}Ω_{W [n−1]}, I)^{†} with W [n − 1]/A^{n} = Z[n]/A^{n} understood. Now let U_{α}/V_{α} be a chart of some points in f^{−1}(D[n]) that is away from f^{−1}(B). Then by shrinking U_{α}/V_{α} and Z_{α} if necessary, we can assume that there is a section z_{α} ∈ Γ(O_{Z_{α}}) so that z_{α} = 0 is the divisor Z_{α} ∩ D[n]. We define Hom_{U_{α}}(f^{∗}Ω_{Z[n]}, I)^{†} be the subgroup of

(φ, η) ∈ Hom_{U_{α}}(f^{∗}Ω_{Z[n]}, I_{α}) ⊕ I_{α}

so that

φ(f^{∗}dz_{α}) = f_{α}^{∗}(z_{α}) · η and φ(dt_{l}) ∈ I_{α}, ∀ l.

Using the A-modules Hom_{U_{α}}(f^{∗}Ω_{Z[n]}, I)^{†}, we can form the complex (D(I)^{•}, ∂), just as we did for W [n] before (1.19). As before, the element 1 ∈ F^{1} ⊗_{A} F^{1∨} defines a flat extension of X /S to X̃/S̃, where S̃ ≜ Spec A ∗ F^{1∨}, with extended sections q̃_{i}, p̃_{j} : S̃ → X̃. We let Ũ_{α}/Ṽ_{α} be the minimal extension of U_{α}/V_{α} as an étale neighborhood of X̃/S̃. We then pick ζ_{α}: Ũ_{α} → Z[n] that is an extension of f_{α}: U_{α} → Z[n] so that ζ_{α} is predeformable and

ζ_{α}^{−1}(D[n]) = ∑_{j=1}^{r} μ_{j}q̃_{j}(Ṽ_{α}). (1.22)

To construct the corresponding complex E(I)^{•} we need two homomorphisms

ζ_{α}(·) : F^{0} ⟶ Hom_{U_{α}}(f^{∗}Ω_{Z[n]}, A)^{†} (1.23)

and

ζ_{αβ}(·) : F^{1} ⟶ Hom_{U_{αβ}}(f^{∗}Ω_{Z[n]}, A)^{†}. (1.24)

First, as before we argue that the difference of ζ_{α} and ζ_{β} over Ũ_{αβ} canonically defines an element ζ_{αβ} ∈ Hom_{U_{αβ}}(f^{∗}Ω_{Z[n]}, F^{1∨})^{†}, which naturally defines a homomorphism as required in (1.24). Here the log differential, namely f^{∗}(dz_{α})/f^{∗}(z_{α}) ↦ η, appear near D[n] because of the constraint (1.22). The construction of (1.23) is similar. Namely, locally over Spec A ∗ F^{0∨} there are two extensions of f: one given by the pullback of ζ_{α} and the other given by the pullback of f. Their difference then gives rise to the homomorphism (1.23).

Once all such data are constructed, we then go ahead to form the homomorphism of complexes F^{•} ⊗_{A} I → D(I)^{•}, form a new complex E(I)^{•} and check that there is a complex of finite rank free A-modules E^{•} = [E^{1} → E^{2}] so that it is quasi-isomorphic to E^{•}, parallel to the argument in the previous subsection.

We now state the main result of this section.

Lemma 1.22. Let the notation be as before and let Λ be a sufficiently fine covering of f by charts of the first or the second kinds. Then the complex E^{•} is a complex of flat A-modules. Further, for any A-module I we have E(I)^{•} = E^{•} ⊗_{A} I and h^{i}(E^{•} ⊗ I) = 0 for i ≠ 1, 2. In particular, there is a complex E^{•} = [E^{1} → E^{2}] of finitely generated free A-modules so that it is quasi-isomorphic to E^{•}.

Theorem 1.23. Let the notation be as in the previous lemma and let S = Spec A be the affine chart of 𝔐(Z[n]^{rel}, Γ)^{st} as before. Then there is a perfect obstruction theory of S taking value in the complex E^{•}. In particular, the functor of the first order deformations 𝔇𝔢𝔣_{S}^{1} is isomorphic to 𝔥^{1}(E^{•}) and there is an obstruction assignment 𝔬𝔟 taking values in 𝔥^{2}(E^{•}) that satisfies the required base change property.

Proof. The proof of the lemma and the theorem are parallel to that of Lemmas 1.15, 1.19 and the Theorem 1.20. The only new ingredient is about preserving the divisor f^{−1}(D[n]) = ∑ μ_{j}q_{j}(S). Since q_{j}(S) ⊂ X is a divisor smooth over S and D[n] ⊂ Z[n] is a smooth divisor, that the deformation of morphisms preserving this relation is given by the sheaf of log differentials is well-known, for example see [20]. Since the proof is routine and parallel to what we did before, we shall omit it. This completes the proof of the theorem. q.e.d.

2. Gromov-Witten invariants

In this section, we will define the virtual moduli cycle of 𝔐(𝔚, Γ), 𝔐(𝔚_{t}, Γ) and 𝔐(ℨ^{rel}, Γ), thus defining the Gromov-Witten invariants of the family W, of the singular variety W_{0} and the relative Gromov-Witten invariants of the pair (Z, D). In the next section, we will prove the decomposition (degeneration) formula relating the Gromov-Witten invariants of W_{t} to the relative Gromov-Witten invariants of the pairs (Y_{1}, D_{1}) and (Y_{2}, D_{2}).

2.1 Perfect obstruction theories of 𝔐(𝔚, Γ) and 𝔐(ℨ^{rel}, Γ).

Recall that the construction of the virtual cycles of moduli stacks is based on the choice of their perfect obstruction theories. In this section, we will show that the perfect obstruction theories constructed in the previous section naturally induce perfect obstruction theories of 𝔐(𝔚, Γ) and 𝔐(ℨ^{rel}, Γ).

Let M be a proper Deligne-Mumford stack with an atlas Λ consisting of finitely many affine étale morphisms ι_{α} : S_{α} = Spec A_{α} → M. We first recall the definition of a perfect obstruction theory of M.

Definition 2.1. A perfect obstruction theory of M (over the atlas Λ) consists of a choice of perfect obstruction theory (E_{α}^{•}, 𝔬𝔟_{α}) of S_{α} for each α ∈ Λ so that they satisfy the following compatibility condition: First, let Ob_{α} be the obstruction sheaf (i.e., = Coker{E_{α}^{1} → E_{α}^{2}}), then the collection {Ob_{α}}_{α∈Λ} descends to a (global) sheaf of O_{M}-modules Ob_{M}. Secondly, the obstruction assignments 𝔬𝔟_{α} and 𝔬𝔟_{β} are identical when pulled back to S_{αβ}, using the given isomorphisms.

Note that {Ob_{α}} descends means that over S_{αβ} the pullback of Ob_{α} and of Ob_{β} are isomorphic and that such isomorphisms satisfy the cocycle condition on S_{αβγ}.

Our immediate goal is to show that the obstruction theories of 𝔐(W [n], Γ)^{st} naturally induces a perfect obstruction theory of 𝔐(𝔚, Γ).

Theorem 2.2. There is a natural perfect obstruction theory of 𝔐(𝔚, Γ) induced by the perfect obstruction theories of 𝔐(W [n], Γ)^{st} constructed in the previous section.

Proof. Let S be an affine chart of 𝔐(𝔚, Γ). Without lose of generality, we can assume S is one of the chart constructed in the proof of Theorem 3.10 in [23]. Namely, there is a chart \overline{S} ⊂ 𝔐(W [n], Γ)^{st} for some n so that S is a closed subscheme of \overline{S} and S → 𝔐(𝔚, Γ) is induced by S → \overline{S} and 𝔐(W [n], Γ)^{st} → 𝔐(𝔚, Γ). We let S = Spec A and \overline{S} = Spec \overline{A}. We then let \overline{E}^{•} = [\overline{E}^{1} → \overline{E}^{2}] be the complex of \overline{A}-modules provided by Theorem 1.20 for the chart \overline{S}.

We begin with the functor of the first order deformations in S. As argued in the proof of [23, Theorem 3.10], there is a neighborhood U of S × {e} ⊂ S × G[n] so that the morphism S × G[n] → 𝔐(W [n], Γ)^{st} induced by the G[n]-action lifts to an étale ϕ: U → \overline{S}. Since U → \overline{S} is étale, each vector v ∈ T_{e}G[n] defines a first order deformation of the inclusion S → \overline{S}, and hence an element \overline{v} ∈ h^{1}(\overline{E}^{•} ⊗\overline{A} A). This induces a homomorphism

T_{e}G[n] ⊗_{𝕜} A ⟶ h^{1}(\overline{E}^{•} ⊗\overline{A} A) ⟶ \overline{E}^{1} ⊗\overline{A} A. (2.1)

Since elements in 𝔐(W [n], Γ)^{st} associate to stable morphisms to 𝔚, at each closed point p ∈ S the homomorphism T_{e}G[n] → T_{p}\overline{S}/T_{p}S induced by the group action is injective. Hence the cokernel of the composite in (2.1) is also free. Now let E^{1} be the cokernel of (2.1) and let E^{2} = \overline{E}^{2} ⊗\overline{A} A. It is direct to check that the composite of (2.1) with \overline{E}^{1} ⊗\overline{A} A → \overline{E}^{2} ⊗\overline{A} A is trivial. Hence \overline{E}^{1} ⊗\overline{A} A → \overline{E}^{2} ⊗\overline{A} A lifts to E^{1} → E^{2}.

We next show that the natural obstruction theory of S takes values in the cohomology theory of E^{•}. First, since (2.1) has free cokernel, E^{•} is a two-term complex of finitely generated free A-modules. Secondly, that the functor of the first order deformations in S is given by the functor 𝔥^{1}(E^{•}) is obvious since the morphism

S × G[n] ⊃ U ⟶^{ϕ} \overline{S} (2.2)

induced by the group action is étale near S ×{e}. Finally, let (B, I, φ) ∈ Ob(𝔗𝔯𝔦_{S}), then it is also an object in 𝔗𝔯𝔦\overline{S} and thus has an obstruction class

𝔬𝔟(B, I, φ) ∈ h^{2}(\overline{E}^{•} ⊗\overline{A} I) = h^{2}(E^{•} ⊗_{A} I)

to extending φ:Spec B/I → S to Spec B → \overline{S}. Because G[n] is smooth and ϕ : U → \overline{S} (in (2.2)) is étale, 𝔬𝔟((B, I, φ)) is also an obstruction class to extending φ to Spec B → S.

Now let S_{α} be charts of 𝔐(𝔚, Γ) with E_{α}^{•} their complexes that are part of their obstruction theories. Then it is direct to check that the collection {h^{2}(E_{α}^{•})} form a sheaf over 𝔐(𝔚, Γ), and the obstruction assignments 𝔬𝔟_{α} are compatible. This completes the proof of Theorem 2.2.

q.e.d.

We now state the theorem concerning the obstruction theory of 𝔐(ℨ^{rel}, Γ).

Theorem 2.3. The perfect obstruction theory of 𝔐(Z[n]^{rel}, Γ)^{st} constructed in the previous section naturally induces a perfect obstruction theory of 𝔐(ℨ^{rel}, Γ).

Proof. We will omit the proof here since it is exactly the same as the proof of the previous theorem. q.e.d.

The next issue is about the obstruction theory of the substack 𝔐(𝔚_{t}, Γ) defined by the fiber product

𝔐(𝔚_{t}, Γ) = 𝔐(𝔚, Γ) ×_{C} t,

where t ∈ C is a closed point. Clearly, when t ≠ 0 the stack 𝔐(𝔚_{t}, Γ) is naturally isomorphic to the module stack of stable morphisms to W_{t} of topological type Γ, which itself admits a natural obstruction theory as worked out in [25]. The obstruction theory of 𝔐(𝔚_{0}, Γ) deserves more attention since it was not known before.

We now study the obstruction theory of 𝔐(𝔚_{0}, Γ). Let W_{0}[n] = W [n] ×_{C} 0. As in the case of 𝔐(𝔚, Γ), we only need to work out the obstruction theory of

𝔐(W_{0}[n], Γ)^{st} ≜ 𝔐(W [n], Γ)^{st} ×_{C} 0.

Let S = Spec A be an affine chart of 𝔐(W [n], Γ)^{st}. Then S_{0} = S ×_{C} 0 is an affine chart of 𝔐(W_{0}[n], Γ)^{st}. We let A_{0} be the quotient ring of A so that S_{0} = Spec A_{0}. As before we let f : X → W [n] be the universal family over S and let f_{0} :X_{0} → W_{0}[n] be the restriction of f to S_{0} ⊂ S. We fix a sufficiently fine covering (U_{α}/V_{α}, f_{α}, W_{α}) of f indexed by Λ and let E^{•} be the associated complex of A-modules constructed in Section 1.1. We let \underline{V_{0}} be the coverings \{V_{0,α}\} of S_{0} with V_{0,α} = V_{α}×_{C} 0. Similarly we let U_{0,α} = U_{α} ×_{C} 0 and let f_{0,α} : U_{0,α} → W_{0}[n] be the restriction of f to U_{0,α}. For any A_{0}-module I, we define

Γ_{V_{0,α_{1}...αm}} (I) ≜ I ⊗_{A_{0}} Γ(O_{V_{0,α_{1}...αm}}). (2.3)

Using these, we can form a Čech complex C^{•}(\underline{V_{0}}, A_{0}) with the standard coboundary operation. Let E_{0}^{•} = E^{•} ⊗_{A} A_{0}. We next construct a homomorphism

\overline{δ}_{i} : E_{0}^{i} → C^{i−1}(\underline{V_{0}}, A_{0}) ⊗_{𝕜} T_{0}C.

Here we understand C^{−1}(\underline{V_{0}}, A_{0}) = 0. Let ξ ∈ E_{0}^{1} be any element. We write ξ = (a, b) with a ∈ F^{1} ⊗_{A} A_{0} and b ∈ C^{0}(Λ, Hom(f^{∗}Ω_{W [n]}, A_{0})^{†}), as in Section 1. Then by the construction in Section 1.1, to each α ∈ Λ the element a defines an extension of U_{0,α}/V_{0,α} to Ũ_{0,α}/Ṽ_{0,α} by A_{0,α} (= Γ_{V_{0,α}}(A_{0})). The extension ζ_{α} chosen before (1.18) induces an extension ζ̃_{0,α}: Ũ_{0,α} → W [n] of f_{0,α}: U_{0,α} → W_{0}[n]. We let π_{n}: W [n] → C be the tautological projection. We now consider π_{n} ∘ ζ̃_{0,α} : Ũ_{0,α} → C. Since π_{n} ∘ ζ̃_{0,α}|_{U_{0,α}} factor through 0 ∈ C,

d(π_{n} ∘ ζ̃_{0,α} − 0) ∈ Γ_{U_{0,α}}(A_{0,α}) ⊗_{𝕜} T_{0}C, A_{0,α} = A_{0} ⊗_{A_{0}} O_{U_{0,α}}.

Because ζ̃_{0,α} is a predeformable extension, the above element lies in A_{0,α} ⊗_{𝕜} T_{0}C. We define \overline{δ}_{1}(a)_{α} to be this element. For \overline{δ}_{1}(b_{α}), since b_{α} ∈ Γ(U_{α}, Hom(f^{∗}Ω_{W [n]}, A_{0})^{†}), b_{α} induces a homomorphism (π_{n} ∘ f)^{∗} T_{0}^{∨}C → A_{0,α}. Again this is an element in A_{0,α} ⊗_{𝕜} T_{0}C. We define \overline{δ}_{1}(b_{α}) to be this element. Clearly, this construction carries over to the case of multi-indices. This defines a map (a homomorphism) of complexes

\overline{δ} : E_{0}^{•} ⟶ C^{•−1}(\underline{V_{0}}, A_{0}) ⊗_{𝕜} T_{0}C.

It is direct to check that this is a homomorphism of complexes. We let F^{•} be the associated complex defined by

F^{i} = E_{0}^{i} ⊕ C^{i−2}(\underline{V_{0}}, A)_{0} ⊗_{𝕜} T_{0}C

whose differential is the obvious induced one. Then we have a short exact sequence of complexes

0 ⟹ C^{•−1}(\underline{V_{0}}, A_{0}) ⊗_{𝕜} T_{0}C ⟹ F^{•} ⟹ E^{•} ⊗_{A} A_{0} ⟹ 0

which induces a long exact sequence of cohomologies for any A_{0}-module I:

0 ⟶ h^{1}(F^{•} ⊗_{A_{0}} I) ⟶ h^{1}(E_{0}^{•} ⊗_{A_{0}} I) ⟶ I ⊗_{𝕜} T_{0}C ⟶

⟶ h^{2}(F^{•} ⊗_{A_{0}} I) ⟶ h^{2}(E_{0}^{•} ⊗_{A_{0}} I) ⟶ 0.

Here we used the fact that since \underline{V_{0}} is an étale covering of S_{0}, h^{j}(C^{•}(\underline{V_{0}}, I)) = I when j = 0 and vanishes when j ≥ 1.

Since terms in C^{•}(\underline{V_{0}}, A_{0}) are flat A_{0}-modules, we can pick a complex of finitely generated free A_{0}-modules F^{•} = [F^{1} → F^{2}] so that F^{•} is quasi-isomorphic to F^{•}.

Proposition 2.4. The chart S_{0} admits a natural perfect obstruction theory taking values in the cohomology of the complex F^{•}.

Proof. We need to check that the functor of the first order deformations 𝔇𝔢𝔣_{S_{0}}^{1} is isomorphic to the functor 𝔥^{1}(F^{•}) and that there is an obstruction assignment taking values in h^{2}(F^{•}) that satisfies the required base change property.

The fact that the functor 𝔇𝔢𝔣_{S_{0}}^{1} is isomorphic to 𝔥^{1}(F^{•}) ≡ 𝔥^{1}(F_{0}^{•}) follows directly from the definition and will be omitted. Now we construct the obstruction assignment. Let (B, I, φ) be any object in 𝔗𝔯𝔦_{S_{0}}. Let T = Spec B/I and f_{T} : X_{T} → W_{0}[n] be the pullback family under φ : T → S_{0}. Let U_{T,α}/V_{T,α} be the pull back of U_{0,α}/V_{0,α} and let f_{T,α}:U_{T,α} → W_{0}[n] be the restriction of f_{T} to U_{T,α}. Recall that in constructing the obstruction class to extending T → S_{0} to T̃ = Spec B → S, we first extend X_{T} /T to XT̃/T̃ and extend f_{T,α} : U_{T,α} → W_{0}[n] to f_{T̃,α} : U_{T̃,α} → W [n], where U_{T̃,α}/V_{T̃,α} is the minimal extension of U_{T,α}/V_{T,α} in XT̃/T̃. We then use the difference of f_{T̃,α} and f_{T̃,β} to build a cocycle a ∈ E ⊗_{A_{0}} I. Let π_{n}:W [n] → C be the tautological projection as before. Since π_{n} ∘ f_{T,α} factor through 0 ∈ C, π_{n} ∘ f_{T̃,α} ∈ I_{α}. Further because f_{T̃,α} is a predeformable extension, it lies in I_{α}. Hence the collection \{π_{n} ∘ f_{T̃,α}\} defines a cochain c ∈ C^{0}(\underline{V_{0}}, I). It is routine to check that the pair (a, c) ∈ F^{2} ⊗_{A_{0}} I is closed, and hence defines a cohomology class [(a, c)] ∈ h^{2}(F^{•} ⊗_{A_{0}} I). Further, it is routine to check that this class is independent of the choice of the extensions f_{T̃,α}, and that it is an obstruction class to extending φ: T → S_{0} to T̃ → S_{0}. We define 𝔬𝔟_{0} be the assignment that assigns (B, I, φ) ∈ Ob(𝔗𝔯𝔦_{S_{0}}) to this class in h^{2}(F^{•} ⊗_{A_{0}} I) = h^{2}(F^{•} ⊗_{A_{0}} I). q.e.d.

Theorem 2.5. The obstruction theories of the charts S_{0} so defined induce a perfect obstruction theory of 𝔐(W_{0}[n], Γ)^{st}, which induces a perfect obstruction theory of 𝔐(𝔚_{0}, Γ).

Proof. The proof is similar to that of Theorem 2.2 and will be omitted. q.e.d.

We comment that so far all the results concerning 𝔚_{0}, including its construction, are based on the existence of the smoothing W of W_{0}. It is not difficult to see that we can construct W_{0}[n] from W_{0} directly, assuming N_{D_{1}/Y_{1}} ≅ N_{D_{2}/Y_{2}}^{∨}. Therefore, we can define 𝔚_{0} and the moduli stack 𝔐(𝔚_{0}, Γ) directly without assuming the existence of W. The construction of the perfect obstruction theory of 𝔐(𝔚_{0}, Γ) without using W is a little tricky, but should be doable. Since we will not use this in this paper, we will content with assuming the existence of a smoothing W of W_{0}.

2.2 Gromov-Witten invariants

The goal of this subsection is to construct the virtual moduli cycles of the moduli stacks 𝔐(𝔚, Γ), 𝔐(𝔚_{t}, Γ) and 𝔐(ℨ^{rel}, Γ) and to define their respective Gromov-Witten invariants.

Currently, there are two constructions of virtual moduli cycles in algebraic geometry. One is the original construction by Tian and the author. They assumed that the moduli space admits a perfect obstruction theory. They then constructed a global cone that function as a virtual normal cone. Such cone was constructed using the (algebraic) Kuranishi maps of the obstruction theory [25, 26]. In their construction they made a technical assumption that there is a global vector bundle on the moduli space that makes the obstruction sheaf its quotient. With this vector bundle, the cone becomes a subcone of this vector bundle, and the virtual moduli cycle is the intersection of this cone with the zero section of this vector bundle, using Gysin map. The alternative construction of Behrend and Fantechi [2, 3] works along a parallel line. They constructed a similar cone as an Artin stack, assuming the moduli space admits a perfect obstruction theory. They then obtain a cone cycle in a vector bundle by assuming the existence of a global vector bundle, as in the original construction of Tian and the author. These two constructions yield identical cycles [19]. Recently, by working out the intersection theory on Artin stacks, Kresh [22] showed that one can construct the virtual moduli cycle without relying on the existence of a global vector bundle as mentioned, thus removing this technical condition. This makes the construction of virtual moduli cycles more versatile. After seeing Kresh’s work, we realized that by applying a simple trick we can remove the technical condition of the existence of such vector bundles in our construction of the virtual moduli cycles. In the following, we will present this modified construction.

We begin with the general situations. Let M be a proper and separated DM-stack. We let Ob be a sheaf of O_{M}-modules. We assume that there is a finite collection of schemes S_{α} and smooth morphisms ρ_{α} : S_{α} → M, indexed by a set Λ, so that the collection of images ρ_{α}(S_{α}) ⊂ M form an open covering of M. We next assume that to each α ∈ Λ there is a locally free sheaf of O_{S_{α}}-modules E_{α}, a surjective homomorphism of sheaves E_{α} → ρ_{α}^{∗}Ob and a cone cycle C_{α} ∈ Z_{∗}Vect(E_{α}) that satisfy the following cycle consistency criteria. Here we denote by Vect(E_{α}) the vector bundle over S_{α} so that its sheaf of sections is E_{α}. In this paper, by abuse of notation we will view a vector bundle as its total space. We first fix a few notations before we state the criteria. Let p ∈ M be any closed point. We pick an étale morphism φ: (X, \overline{p}) → (M, p) and let X̂_{p} be the formal completion of X along \overline{p}. We let G_{p} be the automorphism group of p ∈ M. Note that G_{p} acts naturally on X̂_{p} and up to G_{p} the scheme X̂_{p} is canonical. We next let V\overline{p} = φ^{∗}Ob ⊗_{O_{X}} 𝕜\overline{p} and let VX̂_{p} be the vector bundle V\overline{p} × X̂_{p} over X̂_{p}. Again G_{p} acts on V_{p} and VX̂_{p}, and up to G_{p} they are canonical.

Cycle consistency criteria. We say the collection C = {(S_{α}, E_{α}, C_{α})}_{Λ} satisfies the cycle consistency criteria at p ∈ M if there is a cycle C_{p} ∈ Z_{∗}VX̂_{p} invariant under G_{p} of which the following hold. Let α ∈ Λ be any index, let S_{α,p} = S_{α} ×_{M} p and let Ŝ_{α} = S_{α} ×_{M} X̂_{p}. We let pr_{i} be the i-th projection of the product Ŝ_{α} = S_{α} ×_{M} X̂_{p}. Then there is a surjective homomorphism of vector bundles

Φ_{1} : Vect(E_{α}) ×_{S_{α}} Ŝ_{α} ⟶ VX̂_{p} ×X̂_{p} Ŝ_{α}

extending the canonical (composite) homomorphism^{10}

Vect(E_{α}) ×_{S_{α}} S_{α,p} ⟶ Vect(pr_{1}^{∗}ρ_{α}^{∗}Ob|_{S_{α,p}}) ≡ Vect(pr_{2}^{∗}φ^{∗}Ob|_{S_{α,p}})
≡ V\overline{p} × S_{α,p}

so that Φ_{1}^{∗}C\overline{ξ} = Φ_{2}^{∗}C_{α}. Here the first arrow above is induced by E_{α} → p_{α}^{∗}Ob, Φ_{2} is the tautological flat morphism Vect(E_{α})×_{S_{α}} Ŝ_{α} → Vect(E_{α}) and Φ_{i}^{∗} are the flat pullback homomorphism of cycles.

We will call the collection Ĉ ≜ {C_{p} ⊂ VX̂_{p}}_{p∈M} satisfying the above criteria the infinitesimal models of the collection C. Accordingly we will call C a local model of Ĉ. In the following, we say the collection C is consistent if there is a Ĉ as above that satisfy the above criteria. Conversely, given Ĉ, we say it can be algebraicized if there is a C so that they satisfy the above criteria. Note that once the infinitesimal models exist, then the property of the local model C is completely determined by the infinitesimal models. This is the key to many of the results concerning virtual moduli cycles.

Given a consistent collection C = {(S_{α}, E_{α}, C_{α})}_{Λ} over (M, Ob) we now construct a canonical cycle [C] ∈ A_{∗}M as follows. For each α we let Ξ_{α} be the set of irreducible components of C_{α}. For a ∈ Ξ_{α} we denote by N_{a} the irreducible variety (component) in C_{α} associated to a and let m_{a} be the multiplicity of N_{a} in C_{α}. Then we have

C_{α} = ∑_{a∈Ξ_{α}} m_{a} N_{a} ∈ Z_{∗}Vect(E_{α}). (2.4)

For any a ∈ Ξ_{α} we define the base stack of a to be the minimal closed integral substack Y_{a} ⊂ M so that the natural N_{a} → M factor through Y_{a} ⊂ M. We let j_{a}: Y_{a}^{0} → Y_{a} be the (maximal) dense open substack so that the pullback sheaf j_{a}^{∗}Ob is locally free. Then F_{a}^{0} ≜ Vect(j_{a}^{∗}Ob) is a vector bundle stack over Y_{a}^{0}. Further, the natural morphism

(2.5) η_{a} : Vect(E_{α})|_{ρ_{α}^{−1}(Y_{a}^{0})} ⟶ F_{a}^{0} ≡ Vect(j_{a}^{∗}Ob) induced by E_{α} → ρ_{α}^{∗}Ob is flat. We let N_{a}^{0} ⊂ F_{a}^{0} be the image stack of N_{a}|_{ρ_{α}^{−1}(Y_{a}^{0})} under η_{a} with the reduced stack structure. Clearly, by cycle consistency criteria the flat pullback η_{a}^{∗}N_{a}^{0} contains N_{a}|_{ρ_{α}^{−1}(Y_{a}^{0})} as one of its irrducible components. In the following, we will call N_{a}^{0} ⊂ F_{a}^{0} the intrinsic representative of a. Note that since we choose Y_{a}^{0} to be the maximal possible open substack of Y_{a} so that j_{a}^{∗}Ob is locally free, the open Y_{a}^{0} ⊂ Y_{a} and the substack N_{a}^{0} ⊂ F_{a}^{0} only depend on a.

10For sheaves F of O_{Z}-modules and closed subscheme X ⊂ Z we use F|_{X} to denote F ⊗_{O_{Z}} O_{X}.

Now let b ∈ Ξ_{β} be any element with Y_{b} and N_{b}^{0} ⊂ F_{b}^{0} its base stack and intrinsic representative. We say a ∼ b if Y_{a} = Y_{b} and N_{a}^{0} = N_{b}^{0} in F_{a}^{0} ≡ F_{b}^{0}. This defines an equivalence relation ∼ on ∪_{α∈Λ}Ξ_{α}. We define Ξ = (⋃_{α∈Λ} Ξ_{α})/ ∼. Again by the cycle consistency criteria whenever a ∼ b then m_{a} = m_{b}. Hence each a ∈ Ξ has an associated multiplicity m_{a}, a base substack Y_{a} and an intrinsic representative N_{a}^{0} ⊂ F_{a}^{0} over an open substack Y_{a}^{0} ⊂ Y_{a}.

Assuming there is a global locally free sheaf E on M making Ob its quotient sheaf, then over each Y_{a}^{0} we have a flat projection Vect(E)|_{Y_{a}^{0}} → F_{a}^{0}. We let N_{a} ⊂ Vect(E) be the closure of the pullback of N_{a}^{0} ⊂ F_{a}^{0} under this projection. The associated cycle [C] is then defined to be

[C] = ∑_{a∈Ξ} m_{a}0_{Vect(E)}^{!}[N_{a}] ∈ A_{∗}M, (2.6)

where 0_{Vect(E)}^{!} is the Gysin map of the 0-section of Vect(E). This is essentially the original construction of Tian and the author.

We now back to the general situation (without assuming the existence of such E). We need to define a map ξ:Ξ → A_{∗}M so that ξ(a) is the cycle 0_{Vect(E)}^{!}[N_{a}] should a global E exist. Let a ∈ Ξ be any element. Since M can be covered by a quasi-projective scheme, there is a normal projective variety Y_{a} and a generically finite surjective morphism φ_{a}: Y_{a} → Y_{a}. By abuse of notation, we also view φ_{a} as the composite of Y_{a} → Y_{a} with Y_{a} → M. Since Y_{a} is projective, there is a locally free sheaf of O_{Y_{a}}-modules F_{a} so that φ_{a}^{∗}Ob is a quotient sheaf of F_{a}. We denote by F_{a} the vector bundle Vect(F_{a}) over Y_{a}. Let Y_{a}^{0} ⊂ Y_{a} be a dense open subset so that Y_{a}^{0} → Y_{a}^{0} is étale. Then the morphism h_{a} : F_{a}|_{Y_{a}^{0}} → F_{a}^{0} induced by F_{a} → φ_{a}^{∗}Ob is a flat morphism. We let N_{a} be the closure in F_{a} of the flat pull-back h_{a}^{∗}N_{a}^{0}. Note that N_{a} only depend on Y_{a} and F_{a} → φ_{a}^{∗}Ob. The cycle N_{a} ⊂ F_{a} will be called a representative of a ∈ Ξ. With N_{a} ⊂ F_{a} chosen, we define

ξ(a) = deg(φ_{a})^{−1}φ_{a∗}0_{F_{a}}^{!}[N_{a}], (2.7) where 0_{F_{a}} is the zero section of F_{a}, 0_{F_{a}}^{!} is the Gysin homomorphism Z_{∗}F_{a} → A_{∗}Y_{a} (of the zero section of 0_{F_{a}}) and φ_{a∗} is the push-forward homomorphism of cycles. The degree deg(φ_{a}) is the degree of the morphism φ_{a}:Y_{a} → Y_{a} defined in [38]. Finally, we define

[C] = ∑_{a∈Ξ} m_{a} ξ(a) ∈ A_{∗}M. (2.8)

Note that this construction coincides with that in (2.6) in case a global locally free sheaf E exists.

We will call this construction the basic construction and call [C] the associated cycle of the collection C.

Lemma 2.6. Let the notation be as before. Then ξ(a) is independent of the choice of Y_{a} and F_{a}.

Proof. Let φ_{a,1} : Y_{a,1} → Y_{a} and φ_{a,2} : Y_{a,2} → Y_{a} be two normal varieties and generically finite dominant morphisms and let N_{a,1} ⊂ F_{a,1} and N_{a,2} ⊂ F_{a,2} be the respective choices of the representatives of a over Y_{a,1} and Y_{a,2}. To prove the lemma it suffices to show that

deg(φ_{a,1})^{−1}φ_{a,1∗}0_{F_{a,1}}^{!}[N_{a,1}] = deg(φ_{a,2})^{−1}φ_{a,2∗}0_{F_{a,2}}^{!}[N_{a,2}].

We let Y_{a} be the normalization of an irreducible component of Y_{a,1} ×_{Y_{a}} Y_{a,2} that is dominant and generically finite over Y_{a}. We let p_{i}:Y_{a} → Y_{a,i} be the projection induced by the i-th projection of Y_{a,1}×_{Y_{a}}Y_{a,2}. We pick a locally free sheaf F_{a} on Y_{a} and surjective homomorphism F_{a} → p_{i}^{∗}F_{a,i} so that the diagram

F_{a} ⟶  p_{1}^{∗}F_{a,1}
↓  ↓
p_{2}^{∗}F_{a,2} ⟶ p_{1}^{∗}φ_{a,1}^{∗}Ob ≡ p_{2}^{∗}φ_{a,2}^{∗}Ob

is commutative. Now let F_{a} = Vect(F_{a}), let U ⊂ Y_{a} be a dense open subset so that the projections U → Y_{a}, U → Y_{a,1} and U → Y_{a,2} are flat. Then the flat pull-back of N_{a}^{0} ⊂ F_{a}^{0} via the induced F_{a}|_{U} → N_{a}^{0} is identical to the flat pullback of N_{a,i} ⊂ F_{a,i} under the flat morphism F_{a}|_{U} → F_{a,i}. Further, it is direct to check that deg(φ_{a}) = deg(φ_{a,i}) deg(p_{i}). Hence if we let \overline{N}_{a,i} ⊂ F_{a,i} ×_{Y_{a,i}} Y_{a} ≜ p_{i}^{∗}F_{a,i} be the closure of the flat pullback of N_{a,i}|_{U_{i}} (under F_{a,i} ×_{Y_{a,i}} U → F_{a,i}), then for i = 1 and 2,

φ_{a,i∗}0_{F_{a,i}}^{!}[N_{a,i}] = deg(p_{i})^{−1}φ_{a∗}0_{p_{i}^{∗}F_{a,i}}^{!}[\overline{N}_{a,i}] = deg(p_{i})^{−1}φ_{a∗}0_{F_{a}}^{!}[N_{a}].

This, combined with the identity about the degrees, proves Lemma 2.6. q.e.d.

We now show how to apply this construction to construct the virtual moduli cycle of a Deligne-Mumford stack M endowed with a perfect obstruction theory. Let {S_{α}}_{α∈Λ} be an atlas of M and let {(E_{α}^{•}, 𝔬𝔟_{α})}_{Λ} be the data associated to the perfect obstruction theory of M as in Definition 2.1. Here E_{α}^{•} = [E_{α}^{1} → E_{α}^{2}] is a complex of finite rank locally free sheaves of O_{S_{α}}-modules. We let Ob_{M} be the sheaf of O_{M}-modules that is the descent of Coker{E_{α}^{1} → E_{α}^{2}}. Following [26, Section 3], to each α we can construct a canonical cone cycle C_{α} ⊂ Vect(E_{α}^{2}), using the relative Kuranishi-maps constructed from the perfect obstruction theory of S_{α}. The technical result proved in [26, Section 3] shows that the collection {S_{α}, E_{α}^{2}, C_{α}} satisfies the cycle consistency criterion. Thus by applying the basic construction just explained we construct a cycle, called the virtual moduli cycle of M, associated to the given perfect obstruction theory. We denote this cycle by [M]^{virt}. (For an alternative construction of virtual cycle, see [3, 21].)

By applying this construction to the stacks 𝔐(𝔚, Γ), 𝔐(𝔚_{0}, Γ) and 𝔐(ℨ^{rel}, Γ) with their respective perfect obstruction theories, we obtain the virtual moduli cycles [𝔐(𝔚, Γ)]^{virt}, [𝔐(𝔚_{0}, Γ)]^{virt} and [𝔐(ℨ^{rel}, Γ)]^{virt}. As in [26], for t ∈ C we define the GW-invariant of W_{t} to be the homomorphism

Ψ_{Γ}^{W_{t}} : H^{∗}(W_{t})^{×k} × H^{∗}(𝔐_{g,k}) ⟶ H_{0}(pt) ≅ ℚ

defined by

Ψ_{Γ}^{W_{t}}(α, β) = q_{∗0}(ev^{∗}(α) ∪ π_{g,k}^{∗}(β) [𝔐(𝔚_{t}, Γ)]^{virt}),

where π_{g,k} and ev are the forgetful and evaluation morphisms, g and k are the genus and the number of marked points of the topological type Γ, q : 𝔐(𝔚_{t}, Γ) → {t} ⊂ C is the projection and q_{∗0} is the pushforward A_{∗}𝔐(𝔚_{t}, Γ) → H_{0}(pt) at degree 0. (Here we use H_{∗} to denote the ordinary homology theory in case the ground field is ℂ. Otherwise one can use Chow rings to define the GW-invariants.)

The Gromov-Witten invariants of 𝔚 is the homomorphism

Ψ_{Γ}^{W/C} : H_{C}^{0}(R^{∗}π_{∗}ℚ_{W})^{×k} × H^{∗}(𝔐_{g,k}) ⟶ H_{2}^{BM}(C) ≅ ℚ

defined via a similar formula with q_{∗0} replaced by q_{∗1} : A_{∗}𝔐(𝔚, Γ) → H_{2}^{BM}(C). Here ℚ_{W} is the sheaf of locally constant functions on W taking values in ℚ, π: W → C is the tautological projection and H_{2}^{BM} is the Borel-Moore homology of the open complex curve C.

Now we define the relative Gromov-Witten invariants of Z^{rel} = (Z, D). Let Γ be an admissible weighted graph mentioned before. As explained, it determines the topological type of relative morphisms to ℨ^{rel}. We let Γ^{o} be the sub-data (in Γ) relating to the domain curves. (Namely, the connected components, the genus and both kinds of marked points of the domain curves.) Let 𝔐_{Γ^{o}} be the moduli space of stable curves with topological types Γ^{o}. Here a curve C of topological type Γ^{o} is stable if all connected components of C are stable pointed curves. Clearly, 𝔐_{Γ^{o}} is a Deligne-Mumford stack. As in the ordinary case, there is a forgetful morphism π_{Γ^{o}} : 𝔐(ℨ^{rel}, Γ), Γ) ⟶ 𝔐_{Γ^{o}}. We define the relative GW-invariants to be the homomorphism

Ψ_{Γ}^{Z^{rel}} : H^{∗}(Z)^{×k} × H^{∗}(𝔐_{Γ^{o}}) ⟶ H_{∗}(D^{r})

(recall k and r are the numbers of legs and roots of Γ) defined by

Ψ_{Γ}^{Z^{rel}}(α, β) = q_{∗}(ev^{∗}(α) ∪ π_{Γ^{o}}^{∗}(β) [𝔐(ℨ^{rel}, Γ)]^{virt}) ∈ H_{∗}(D^{r}).

Here

(2.9) q : 𝔐(ℨ^{rel}, Γ) → D^{r} and ev : 𝔐(ℨ^{rel}, Γ) ⟶ Z^{k}

are the morphisms defined by evaluating on the distinguished and the ordinary marked sections respectively.

3. Degenerations of Gromov-Witten invariants

In this and the next section, we will prove the degeneration formula of the Gromov-Witten invariants of the family W/C stated in the introduction of this paper. We will state the first version of the degeneration formula in the first subsection. We will state the reduction lemmas in Subsection 3.2. The proof of these lemmas will be given in the next section.

3.1 The first version of the degeneration formula

The first step to prove the degeneration formula is to express 𝔐(𝔚_{0}, Γ) as a union of Cartier-divisors in 𝔐(𝔚, Γ). Here we fix a Γ = (g, k, b) once and for all. We first define the notion of Cartier-divisor of an algebraic stack M.

Definition 3.1. Let M be an algebraic stack. A C-divisor on M is a pair (L, s) where L is a line bundle on M and s is a section of L. An isomorphism between (L, s) and (L′, s′) consists of an isomorphism L ≅ L′ so that s ≡ s′ under this isomorphism.

We comment that a C-divisor (L, s) over a scheme is a pseudo-divisor (defined in [7]) via (L, Z, s) where Z = X − s^{−1}(0). Note that when (L, s) and (L′, s′) are two C-divisors, then (L, s)⊗(L′, s′) ≜ (L⊗L′, ss′) is also a C-divisor. Also, we do not require that s be nontrivial.

We now let Ω be the set of all admissible triples defined in [23, Section 4] and reviewed in the introduction. Recall that η = (Γ_{1}, Γ_{2}, I) ∈ Ω is an admissible triple if Γ_{1} and Γ_{2} are two weighted graphs of identical numbers of roots and I is an order preserving inclusion I : [k_{1}] → [k] where k_{i} is the number of legs of Γ_{i} and k = k_{1} + k_{2}. Let C_{1} and C_{2} be two curves of topological types Γ_{1} and Γ_{2}, respectively. We can identify the i-th distinguished marked point q_{1,i} ∈ C_{1} with the i-th distinguished marked point q_{2,i} ∈ C_{2} for all i to obtain a new curve C ≜ C_{1} ⊔ C_{2}. It has k marked points, ordered according to I. As part of the definition of Ω, we require that the multiplicities of the i-th roots of Γ_{1} and Γ_{2} are identical, that C is connected of genus g and b = ∑_{x∈V (Γ_{1})∪V (Γ_{2})} b(x) (b(x) accounts for the degree of the stable morphism along the connected component labeled by x). We will call C the gluing of C_{1} and C_{2} along distinguised marked points. This gluing construction can be applied to a pair of families of curves. Hence for each η = (Γ_{1}, Γ_{2}, I) ∈ Ω, we have a closed local immersion of stacks

(3.1) Φ_{η} : 𝔐(𝔜_{1}^{rel}, Γ_{1}) ×_{D^{r}} 𝔐(𝔜_{2}^{rel}, Γ_{2}) ⟶ 𝔐(𝔚, Γ),

defined in [23, (4.8)]. Here 𝔐(𝔜_{i}^{rel}, Γ_{i}) → D^{r} is the evaluation morphism q in (2.9) and the morphism (3.1) is defined by sending any pair ((f_{1}, X_{1}), (f_{2}, X_{2})) to the family (f_{1} ⊔ f_{2}, X_{1} ⊔ X_{2}). Following [23], we define 𝔐(𝔜_{1}^{rel} ⊔ 𝔜_{2}^{rel}, η) be the image stack of (3.1). As was shown in [23, Section 4],

(3.2) 𝔐(𝔜_{1}^{rel}, Γ_{1}) ×_{D^{r}} 𝔐(𝔜_{1}^{rel}, Γ_{1}) ⟶ 𝔐(𝔜_{1}^{rel} ⊔ 𝔜_{2}^{rel}, η)

is finite, étale of pure degree | Eq(η)|. Here by abuse of notion we also use Φ_{η} to denote this induced morphism.

In this subsection, to each η ∈ Ω we will define a C-divisor (L_{η}, s_{η}) on 𝔐(𝔚, Γ) so that the vanishing locus (as topological space) of s_{η} is 𝔐(𝔜_{1}^{rel} ⊔𝔜_{2}^{rel}, η). We begin with the study of line bundles on A^{n+1}. We continue to use the convention introduced in [23, Section 1] concerning subsets of A^{n+1}. For l ∈ [n + 1] we denote by H_{l} ⊂ A^{n+1} the l-th coordinate hyperplane of A^{n+1}. We define (s_{l}, L_{l}) be a pair of a section of a line bundle on A^{n+1} so that O_{A^{n+1}}(L_{l}) ≅ O_{A^{n+1}}(H_{l}) and the section s_{l} is the constant section 1 ∈ O_{A^{n+1}} under this isomorphism. Recall that A^{n+1} is a G[n]-subscheme (G[n] ≜ GL(1)^{×n}) as defined in [23, Section 1]. Since H_{l} ⊂ A^{n+1} is invariant under the G[n]-action, there is a unique G[n]-linearization on L_{l} so that the section s_{l} is G[n]-invariant. We fix such a linearization. Now let J : [m + 1] → [n + 1] be an order preserving embedding. Following the convention in [23, Section 1], J defines a standard embedding^{11} γ_{J}:A^{m+1} → A^{n+1} and hence defines a pullback C-divisor γ_{J}^{∗}(L_{l}, s_{l}) on A^{m+1}. There are two possibilities: One is when l ≠ Im(J). Then Im(γ_{J})∩H_{l} = ∅ and hence there is a canonical isomorphism^{12} γ_{J}^{∗}(L_{l}, s_{l}) ≅ (1_{A^{m+1}}, 1). The other case is when J(l′) = l for some l′ ∈ [m + 1], in which case we have γ_{J}^{∗}(L_{l}, s_{l}) ≅ (L_{l′}, s_{l′}).

Now let J:[n_{1} +1] → [n_{2} +1] be an order preserving embedding. Let S be any scheme, τ : S → A^{n_{1}+1} and ρ: S → G[n_{2}] be two morphisms with γ_{J}:A^{n_{1}+1} → A^{n_{2}+1} the standard embedding. As in [23], we define (γ_{J} ∘ τ)^{ρ}: S → A^{n_{2}+1} be the morphisms induced by γ_{J} ∘ τ : S → A^{n_{2}+1} and the G[n_{2}]-action on A^{n_{2}+1} via ρ.

Lemma 3.2. Let J, τ and ρ be as before. In case l_{2} = J(l_{1}) then we have a natural isomorphism ((γ_{J} ∘ τ)^{ρ})^{∗}(L_{l_{2}}, s_{l_{2}}) = τ^{∗}(L_{l_{1}}, s_{l_{1}}). In case l_{2} ∉ Im(J) the same identity holds with (L_{l_{1}}, s_{l_{1}}) replaced by (1, 1).

Proof. We have the canonical isomorphism (γ_{J} ∘ τ)^{∗}(L_{l_{2}}, s_{l_{2}}) ≅ τ^{∗}(L_{l_{1}}, s_{l_{1}}). The required isomorphism is then induced by the canonical isomorphism

((γ_{J} ∘ τ)^{ρ})^{∗}(L_{l_{2}}, s_{l_{2}}) ≅ (γ_{J} ∘ τ)^{∗}(L_{l_{2}}, s_{l_{2}})

induced by the G[n_{2}]-linearization on (L_{l_{2}}, s_{l_{2}}). q.e.d.

We now construct the required C-divisor (L_{η}, s_{η}) on 𝔐(𝔚, Γ). Let S → 𝔐(𝔚, Γ) be any chart with f : X → W its universal family. Without loss of generality, we can assume that W = W [n] ×_{C[n]} S via a τ:S → C[n]. We let

S_{η} = S ×_{𝔐(𝔚,Γ)} 𝔐(𝔜_{1}^{rel} ⊔ 𝔜_{2}^{rel}, η). (3.3)

11For instance in case J : [2] → [4] is defined by J(1) = 1 and J(2) = 3, then γ_{J}:A^{2} → A^{4} is defined by γ_{J}(t_{1}, t_{2}) = (t_{1}, 1, t_{2}, 1).

12We use bold 1_{X} with subscription X to denote the trivial line bundle on X.

In case S_{η} = ∅, we define (L_{η}, s_{η})|_{S} be (1_{S}, 1). When S_{η} ≠ ∅, we consider the tautological projection ρ_{η}:S_{η} → 𝔐(𝔜_{1}^{rel} ⊔ 𝔜_{2}^{rel}, η) and the composite p_{i} ∘ ρ_{η}:S_{η} → 𝔐(𝔜_{i}^{rel}, Γ_{i}). Here

p_{i} : 𝔐(𝔜_{1}^{rel} ⊔ 𝔜_{2}^{rel}, η) ⟶ 𝔐(𝔜_{i}^{rel}, Γ_{i})

is the i-th projection, which exists if we replace S by an étale cover of S. By shrinking S if necessary, we can assume that the pullback of the universal family of 𝔐(𝔜_{i}^{rel}, Γ_{i}) to S_{η} (via p_{i} ∘ ρ_{η}) is given by a family f_{i}:X_{i} → Y_{i}, where Y_{i} is an effective relative pair in 𝔜_{i}^{rel}(S_{η}) associated to a morphism τ_{i} : S_{η} → A^{n_{i}}. Following the discussion leading to the proof of Proposition [23, Prop. 4.12], the tautological family over S_{η} of the morphism Φ_{η}|_{S_{η}}:S_{η} → 𝔐(𝔚, Γ) is represented by the family

f_{1} ⊔ f_{2} : X_{1} ⊔ X_{2} ⟶ Y_{1}^{o} ⊔ Y_{2}

with Y_{1}^{o} ⊔ Y_{2} ∈ 𝔚(S_{η}) given by the morphism τ_{η} : S_{η} → C[n], where n = n_{1} + n_{2}, defined in [23, (4.4)]^{13} . By definition f_{1} ⊔ f_{2} is isomorphic to the restriction of f to the family over S_{η}, denoted by f|_{S_{η}}. Namely there are isomorphisms shown below that make the following diagram commutative

\begin{matrix}X_{1} ⊔ X_{2} & ⟶^{f_{1}⊔f_{2}} & Y_{1}^{o} ⊔ Y_{2} \\ ≅↓ & & ≅↓ \\ X ×_{S} S_{η} & ⟶^{f|_{Sη}} & W ×_{S} S_{η}. \end{matrix} (3.4)

Now let D_{1}, …, D_{n+1} be the n + 1 components of the singular locus of the fibers of W [n] over C[n]. For any closed z ∈ S_{η} there is an integer l_{z} ∈ [n + 1] so that the images of the distinguished divisors D_{1,z} ⊂ Y_{1,z} and of D_{2,z} ⊂ Y_{2,z} under the obvious morphism

D_{1,z} ≅ D_{2,z} ⊂ Y_{1}^{o} ⊔ Y_{2} ⟶^{≅} W ×_{S} S_{η} ⟶ W [n]

lie in D_{l_{z}}. Clearly, l_{z} is locally constant on S_{η}. Hence by shrinking S if necessary we can assume that it is constant on S_{η}, say is l_{η} ∈ ℤ. In the following we will call f|_{S_{η}} = f_{1} ⊔ f_{2} the η-decomposition of f and call the divisor D_{l_{η}} ⊂ W [n] the locus where the η-decomposition of f|_{S_{η}} takes place.

13Y_{1}^{o} ⊔ Y_{2} is the result of gluing the distinguished divisors of Y_{1} and Y_{2} in the obvious way.

Definition 3.3. Let S → 𝔐(𝔚, Γ) be a chart with f:X → W the universal family, where W = W [n] ×_{C[n]} S. We say S is η-admissible if there is an integer l ∈ [n + 1] so that the tautological S_{η} → C[n], where S_{η} is defined in (3.3), factor through C[n] ×_{A^{n+1}} H_{l} ⊂ C[n] and that the divisor D_{l} ⊂ W [n] (or the locus X ×_{W [n]} D_{l} ⊂ X) is where the η-decomposition of f|_{S_{η}} takes place.

Clearly for each η ∈ Ω we can find an atlas Λ of 𝔐(𝔚, Γ) so that all its charts are η-admissible. Let Λ be such an atlas and let S_{α} be any chart in this atlas. We let f_{α}: X_{α} → W [n_{α}] ×_{C[n_{α}]} S_{α} be the universal family with τ_{α} : S_{α} → C[n_{α}] the tautological morphism. We let l_{α} be the integer so that D_{l_{α}} ⊂ W [n_{α}] is where the η-decomposition of f_{α}|_{S_{α,η}} takes place. We then define the C-divisor (L_{η,α}, s_{η,α}) on S_{α} to be the pull back of the C-divisor (L_{l_{α}}, s_{l_{α}}) on A^{n_{α}+1} via S ⟶^{τ_{α}} C[n_{α}] → A^{n_{α}+1}. Because the chart S_{α} is η-admissible, the vanishing locus of s_{η,α} is exactly S_{α} ∩ 𝔐(𝔜_{1}^{rel} ⊔ 𝔜_{2}^{rel}, η).

Lemma 3.4. The collection (L_{η,α}, s_{η,α})_{α∈Λ} forms a C-divisor on 𝔐(𝔚, Γ).

Proof. Let S_{α} and S_{β} be two charts in Λ. We consider S_{αβ} = S_{α}×_{𝔐(𝔚,Γ)}S_{β} with its projections ρ_{α}, ρ_{β}. Let f_{α} and f_{β} be the universal families over S_{α} and S_{β} and let ρ_{α}^{∗}(f_{α}) and ρ_{β}^{∗}(f_{β}) be the pullback families. We let the isomorphism of the families ρ_{α}^{∗}(f_{α}) and ρ_{β}^{∗}(f_{β}) be given by the (left hand side of the) diagram

\begin{aligned}X_{α} ×_{S_{α}} S_{αβ} & ⟶^{f_{α}|_{S_{αβ}}} & W_{α} ×_{S_{α}} S_{αβ} & ⟵^{⊃} & D_{l_{α}} ×_{C[n_{α}]} S_{αβ} \\ ^{φ_{1}}↓≅ & & ^{φ_{2}}↓≅ & & ≅↓ \\ X_{β} ×_{S_{β}} S_{αβ} & ⟶^{f_{β}|_{S_{αβ}}} & W_{β} ×_{S_{β}} S_{αβ} & ⟵^{⊃} & D_{l_{β}} ×_{C[n_{β}]} S_{αβ}. \end{aligned} (3.5)

We distinguish two cases: The first is when S_{αβ} is disjoint from 𝔐(𝔜_{1}^{rel}⊔ 𝔜_{2}^{rel}, η). Then the pullbacks of (L_{η,α}, s_{η,α}) and (L_{η,β}, s_{η,β}) to S_{αβ} are canonically isomorphic to the trivial C-divisor (1_{S_{αβ}}, 1), hence they are naturally isomorphic to each other. The other case is when S_{αβ} ∩ 𝔐(𝔜_{1}^{rel} ⊔ 𝔜_{2}^{rel}, η) ≠ ∅. Since S_{α} and S_{β} are η-admissible, they have the associated morphisms τ_{α} : S_{α} → C[n_{α}] and τ_{β} : S_{β} → C[n_{β}] and the associated integers l_{α} and l_{β} respectively. Because of the isomorphisms in (3.5), we have D_{l_{α}} ×_{C[n_{α}]} S_{αβ} ≅ D_{l_{β}} ×_{C[n_{β}]} S_{αβ} as shown in the above diagram. Now let T be any open subset of S_{αβ} so that the restriction of φ_{2} to W_{α} ×_{S_{α}} T ≅ W_{β} ×_{S_{β}} T is induced by a sequence of effective arrows^{14} . Then by Lemma 3.2, the restriction to T of the pullback ρ_{α}^{∗}(L_{η,α}, s_{η,α}) is canonically isomorphic to the restriction to T of ρ_{β}^{∗}(L_{η,β}, s_{η,β}). By [23, Lemma 1.8], we can cover S_{αβ} by such T ’s. Further, applying Lemma 3.2 again we see immediately that the isomorphisms ρ_{α}^{∗}(L_{η,α}, s_{η,α})|_{T} ≅ ρ_{β}^{∗}(L_{η,β}, s_{η,β})|_{T} patch together to form an isomorphism

φ_{βα} : ρ_{α}^{∗}(L_{η,α}, s_{η,α}) ⟶^{≅} ρ_{β}^{∗}(L_{η,β}, s_{η,β}).

Since the isomorphism φ_{αβ} canonically depends on the isomorphism (3.5), over S_{αβγ} we have φ_{αβ} ∘ φ_{βγ} = φ_{αγ}. Thus the collection (L_{η,α}, s_{η,α})_{α∈Λ} coupled with the isomorphisms φ_{αβ} defines a C-divisor on 𝔐(𝔚, Γ). We denote the resulting C-divisor by (L_{η}, s_{η}). q.e.d.

We now indicate the relation between the C-divisor (L_{η}, s_{η}) and the canonical log structure on 𝔐(𝔚, Γ). Recall that the log structure defined in Section 1.1 defines a canonical log structure on 𝔐(W [n], Γ)^{st}, which is G[n]-equivariant and thus descends to a canonical log structure N on 𝔐(𝔚, Γ). The line bundle L_{η} with the section s_{η} defines also a log structure L_{η} on 𝔐(𝔚, Γ). It follows from the construction of N_{α} (see (1.7)) and (L_{η}, s_{η}) that the identity map is a log morphism (𝔐(𝔚, Γ), L_{η}) → (𝔐(𝔚, Γ), N). Using the chart V in (1.7), this is given by the homomorphism of prelog structure ℕ → N_{l} ⊂ N_{V} defined by 1 ↦ m_{α}e_{α} for α ∈ K_{l}.

Associating to each closed point t ∈ C, considered as an effective divisor in C, we have a C-divisor (L_{t}, r_{t}) such that O_{C}(L_{t}) = O_{C}(t) and that r_{t} is the section induced by the constant section 1 ∈ Γ(C, O_{C}) together with the natural homomorphism O_{C}(L_{t}) → O_{C}(t). We let (L_{t}, r_{t}) be the C-divisor on 𝔐(𝔚, Γ) that is the pullback of (L_{t}, r_{t}) via the tautological projection 𝔐(𝔚, Γ) → C. When t = 0 ∈ C we denote the corresponding C-divisor by (L_{0}, r_{0}). Recall that any triple η = (Γ_{1}, Γ_{2}, I) ∈ Ω with r roots is defined to be similar to η^{σ} = (Γ_{1}^{σ}, Γ_{2}^{σ}, I) for any permutation σ ∈ S_{r}. Note that when η_{1} ∼ η_{2} then (L_{η_{1}}, s_{η_{1}}) ≡ (L_{η_{2}}, s_{η_{2}}).

Proposition 3.5. The tensor product of the C-divisors {(L_{η}, s_{η}) | η ∈ Ω/∼} is isomorphic to (L_{0}, r_{0}).

Proof. Let Λ be an atlas of 𝔐(𝔚, Γ) so that all its charts are ηadmissible for all η ∈ Ω/∼. Since Ω/∼ is a finite set, such atlas does exist. Now let S_{α} be any chart in Λ. We let Ω_{α} be those triples η ∈ Ω so that Im(Φ_{η}) ∩ S_{α} ≠ ∅. Since (L_{η,α}, s_{η,α}) ≅ (1_{S_{α}}, 1) canonically when η ∉ Ω_{α}/∼,

14For the definition of effective arrows please see [23, Section 1].

⊗_{η∈Ω/∼}(L_{η}, s_{η})|_{S_{α}} ≡ ⊗_{η∈Ω_{α}/∼}(L_{η,α}, s_{η,α}).

Now let f_{α} : X_{α} → W_{α} be the universal family over S_{α} and let τ_{α} : S_{α} → C[n_{α}] be so that W_{α} = τ_{α}^{∗}W [n_{α}]. To each η ∈ Ω_{α} we let l_{η} be the integer defined in Definition 3.3. The assignment η ↦ l_{η} defines a function Ω_{α}/∼→ [n_{α}+1]. Because of [23, Lemma 4.13], this assignment is one-to-one. Now let K_{α} ⊂ [n_{α} +1] be the image set of this assignment and let \overline{τ}_{α}: S_{α} → A^{n_{α}+1} be the composition of τ_{α} with the projection C[n_{α}] → A^{n_{α}+1}. Clearly, if l ∈ [n_{α} + 1] − K_{α}, then \overline{τ}_{α}(S_{α}) ∩ H_{l} = ∅ and hence \overline{τ}_{α}^{∗}(L_{l}, s_{l}) ≡ (1_{S_{α}}, 1)^{15} . Otherwise, l = l_{η} for a unique η ∈ Ω_{α}/ ∼ and then \overline{τ}_{α}^{∗}(L_{l}, s_{l}) ≡ (L_{η,α}, s_{η,α}). Therefore, we have canonical isomorphisms

⊗_{η∈Ω_{α}/∼}(L_{η,α}, s_{η,α})|_{S_{α}} ≅ ⊗_{l∈K_{α}} \overline{τ}_{α}^{∗}(L_{l}, s_{l}) ⊗ ⊗_{l∈[n_{α}]−A}\overline{τ}_{α}^{∗}(L_{l}, s_{l})
≅ \overline{τ}_{α}^{∗}(⊗_{l∈K_{α}}(L_{l}, s_{l}))
≅ (L_{0}, r_{0})|_{S_{α}}.

Because the above isomorphisms are canonical, they are compatible over S_{αβ} and hence define an isomorphism of C-divisors as required by the proposition. q.e.d.

We now derive the first version of the degeneration formula. We first recall the notion of localized top Chern class of a vector bundle with a section. Let E be a rank m vector bundle over a scheme X and s a section of E. The localized top Chern class of (E, s) is the homomorphism

c_{m}(E, s):A_{∗}X → A_{∗−m}s^{−1}(0)

defined in [7] as follows: Let Z be any variety in X. We take the normal cone N_{s^{−1}(0)∩Z/Z} to s^{−1}(0) ∩ Z in Z and then define c_{m}(E, s)([Z]) = 0_{E}^{!}(N_{s^{−1}(0)∩Z/Z}), where 0_{E}^{!} is the Gysin map of the zero section 0_{E} of E|_{s^{−1}(0)}. This defines a homomorphism of the group of cycles. This construction can be extended to the case where X is an algebraic stack [38] with A_{∗}X understood to be the cycle group with rational coefficients.

15There is an exceptional case I should mention here. It is when there are s ∈ S_{α} so that \overline{τ}_{α}(s) ∈ H_{l} while f_{α}^{−1}(D_{l}) ∩ X_{s} = ∅. Note that since X_{s} is connected, this is possible only when l = 1 or n_{α} + 1. In either case, we agree that X_{s} decomposes into C_{1} ⊔ C_{2} with either C_{1} = ∅ or C_{2} = ∅, and the corresponding η = (Γ_{1}, ∅, ∅) or η = (∅, Γ_{2}, ∅), which we agree is in Ω/∼. In this case we let l_{η} be 1 or n_{α} + 1. With this agreement, this statement is true without exception.

We consider the moduli stack 𝔐(𝔚, Γ) and its Gromov-Witten invariants

Ψ_{Γ}^{W/C} : H_{C}^{0}(R^{∗}π_{∗}ℚ_{W})^{×k} × H^{∗}(𝔐_{g,k}) ⟶ H_{2}^{BM}(C) ≅ ℚ.

Now let ξ ∈ C be any closed point and let H_{C}^{0}(R^{∗}π_{∗}ℚ_{W}) → H^{∗}(W_{ξ}) be induced by W_{ξ} → W. As our convention, for α ∈ H_{C}^{0}(R^{∗}π_{∗}ℚ_{W}) we denote by α(ξ) its image in H^{∗}(W_{ξ}). We let H_{2}^{BM}(C) → ℚ be the Gysin homomorphism defined by intersecting with the divisor ξ ∈ C. Then we have a commutative diagram

\begin{matrix}H_{C}^{0}(R^{∗}π_{∗}ℚ_{W})^{×k} × H^{∗}(𝔐_{g,k}) & ⟶^{Ψ_{Γ}^{W/C}} & H_{2}^{BM}(C) ≅ ℚ \\ ↓ & & ↓ \\ H^{∗}(W_{ξ})^{×k} × H^{∗}(𝔐_{g,k}) & ⟶^{Ψ_{Γ}^{W_{ξ}}} & ℚ. \end{matrix} (3.6)

This was proved in [25] for ξ ≠ 0 except that there we used the existence of a global vector bundle in defining the GW-invariants. This will be proved later in this paper again. We let Γ = (g, k, b) be the triple as before and let α ∈ H^{0}(R^{∗}π_{∗}ℚ_{W})^{×k} and let β ∈ H^{∗}(𝔐_{g,k}) be any classes. As before, we let ev_{ξ} : 𝔐(W_{ξ}, Γ) → W_{ξ}^{×k} be the evaluation morphism by the ordinary marked points of the stable morphisms, and let π_{g,k}:𝔐(𝔚, Γ) → 𝔐_{𝔤,k} be the forgetful map.

Theorem 3.6. For any closed ξ ∈ C, we have

Ψ_{Γ}^{W_{ξ}}(α(ξ), β)

= q_{∗0} ( ∑_{η∈Ω/∼}(ev_{0}^{∗}(α(0)) ∪ π_{g,k}^{∗}(β))(c_{1}(L_{η}, s_{η})[𝔐(𝔚, Γ)]^{virt})) .

Proof. First, by the commutativity of the diagram (3.6), we have Ψ_{Γ}^{W_{ξ}}(α(ξ), β) = Ψ_{Γ}^{W_{0}}(α(0), β). In the later part of this paper, we will show that

Ψ_{Γ}^{W_{0}}(α(0), β) = q_{∗0}((ev_{0}^{∗}(α(0)) ∪ π_{g,k}^{∗}(β))(c_{1}(L_{0}, r_{0})[𝔐(𝔚, Γ)]^{virt})).

Since (L_{0}, r_{0}) is the tensor product of all (L_{η}, s_{η}), the Chern class operations

c_{1}(L_{0}, r_{0}) = ∑_{η∈Ω/∼} c_{1}(L_{η}, s_{η}) : A_{∗}𝔐(𝔚, Γ) → A_{∗}𝔐(𝔚_{0}, Γ).

The theorem then follows immediately. q.e.d.

3.2 Statement of the degeneration formula

In this subsection, we will first construct the virtual moduli cycles of several moduli stacks related to the substack 𝔚_{0} ⊂ 𝔚. After that, we will state the final version of the degeneration formula of the Gromov-Witten invariants of this paper. We will leave the proof of the key lemmas to the next section.

Let η ∈ Ω/∼ be any admissible triple. Associated to η we have the substack 𝔐(𝔜_{1}^{rel} ⊔ 𝔜_{2}^{rel}, η) ⊂ 𝔐(𝔚, Γ) that is the image stack of Φ_{η} in (3.1). Let (L_{η}, s_{η}) be the C-divisor on 𝔐(𝔚, Γ) defined in the previous subsection. We define the substack

𝔐(𝔚_{0}, η) = s_{η}^{−1}(0) ⊂ 𝔐(𝔚, Γ).

Note that we have an increasing chain of closed substacks

𝔐(𝔜_{1}^{rel} ⊔ 𝔜_{2}^{rel}, η) ⊂ 𝔐(𝔚_{0}, η) ⊂ 𝔐(𝔚_{0}, Γ) ⊂ 𝔐(𝔚, Γ),

where the first inclusion induces a homeomorphism on topological spaces. In the previous section, we have constructed the perfect-obstruction theories of 𝔐(𝔚, Γ) and of 𝔐(𝔚_{0}, Γ), and thus have constructed their virtual moduli cycles. In the first part of this subsection, we will show that the natural obstruction theories of 𝔐(𝔜_{1}^{rel} ⊔ 𝔜_{2}^{rel}, η) and of 𝔐(𝔚_{0}, η) are also perfect. Thus they have natural virtual moduli cycles.

We first investigate the obstruction theory of 𝔐(𝔚_{0}, η). The discussion is parallel to the obstruction theory of 𝔐(𝔚_{0}, Γ). We now present the details. Recall that for each n we have the moduli 𝔐(W [n], Γ)^{st} of stable morphisms to W [n] of topological type Γ that are also stable as morphisms to the stack 𝔚. We let 𝔐(W [n], Γ)^{st} → 𝔐(𝔚, Γ) be the tautological morphism and let

𝔐(W_{0}[n], η)^{st} = 𝔐(W [n], Γ)^{st} ×_{𝔐(𝔚,Γ)} 𝔐(𝔚_{0}, η).

As in the case of 𝔐(𝔚, Γ), we first work out the obstruction theory of 𝔐(W_{0}[n], η)^{st}. Of course for étale charts S of 𝔐(W [n], Γ)^{st} we can define the notion of η-admissible as in Definition 3.3. Now let S be an η-admissible chart of 𝔐(W [n], Γ)^{st} and let S_{η} = S ×_{𝔐(W [n],Γ)^{st}} 𝔐(W_{0}[n], η)^{st}. As before, we let f : X → W [n] be the universal family over S. We let l be the integer so that f^{−1}(D_{l}) is where the ηdecomposition of f|_{S_{η}} takes place. We fix a covering of f by charts (U_{α}/V_{α}, f_{α}) of the first and the second kinds, indexed by Λ. Following our convention, we denote by \underline{U} the étale covering \{U_{α}\}_{Λ} of X. We let f_{η,α} : U_{η,α} → W [n] be the restriction of f_{α} to U_{η,α} ≜ U_{α} ×_{S} S_{η}. We let V_{η,α} = V_{α} ×_{S} S_{η} and let \underline{V_{η}} be the covering \{V_{η,α}\}_{Λ} of S_{η}. We let A = Γ(O_{S}) and let E^{•} ≡ E(A)^{•} be the complex constructed in Section 2 associated to the covering Λ. Now let L_{η} be the line bundle on S_{η} that is the pullback of L_{η} via S_{η} → 𝔐(𝔚, Γ). We form the ordinary Čech complex

C_{η}^{•} ≜ C^{•}(\underline{V_{η}}, L_{η}) (3.7)

of the invertible sheaf (line bundle) L_{η} over S_{η} associated to the covering \underline{V_{η}}. Let A_{η} = Γ(O_{S_{η}}). We define a homomorphism of the complex:

δ_{k} : E^{k} ⊗_{A} A_{η} ⟹ C_{η}^{k−1} ≜ C^{k−1}(\underline{V_{η}}, L_{η}) (3.8)

as follows: Let (a, b) ∈ E^{1} ⊗_{A} A_{η} be any element. As argued in the proof of Proposition 1.16, a defines, to each α ∈ Λ, a flat extension Ũ_{η,α}/Ṽ_{η,α} of U_{η,α}/V_{η,α} by the module Γ(O_{V_{η,α}}) and predeformable extensions ζ_{η,α}: Ũ_{η,α} → W [n] of f_{η,α} : U_{η,α} → W [n]. Let ρ_{η,α} : V_{η,α} → A^{n+1} be the tautological morphism (induced by S → A^{n+1}) and let ρ̃_{η,α} : Ṽ_{η,α} → A^{n+1} be the composite of ζ_{η,α} and the projection W [n]_{α} → A^{n+1}. Since t_{l} ∘ ρ_{η,α} = 0,

ξ_{α}(a) ≜ d(t_{l} ∘ ρ̃_{η,α} − 0) ∈ Γ(V_{η,α}, ρ_{η,α}^{∗}N_{H_{l}/A^{n+1}}) ≡ Γ(V_{η,α}, L_{η}),

where N_{H_{l}/A^{n+1}} is the normal bundle to H_{l} in A^{n+1}. Here we have used the fact that ρ_{η,α}^{∗}N_{H_{l}/A^{n+1}} is canonically isomorphic to L_{l}. On the other hand, for b = {b_{α}} with b_{α} ∈ Hom_{U_{α}}(f^{∗}Ω_{W [n]}, A_{η})^{†}, b_{α}(f^{∗}(dt_{l})) ∈ I. We define ξ(b_{α}) = b_{α}(f^{∗}(dt_{l})). This defines the homomorphism δ_{1} via

δ_{1}((a, b))_{α} = ξ_{α}(a) + ξ(b_{α}). (3.9)

Note that for k > 1 elements in E^{k} are of the form b = {b_{α_{0}...α_{k}}} and we can define δ_{k}(b_{α_{0}...α_{k}}) similarly. This defines δ_{k} for k > 1. It is direct to check that the so defined map is a homomorphism of complexes. With this homomorphism of complexes, we can form a complex

E_{η}^{k} ≜ C_{η}^{k−2} ⊕ E^{k} ⊗_{A} A_{η} (3.10)

with the induced differential. It is clear that the statement of Lemma 1.15 holds true to E_{η}^{•}. Namely, for sufficiently fine admissible covering Λ of f, E_{η}^{•} is a complex of flat A_{η}-modules. Further, to each A_{η}-module I we have the following exact sequence of complexes

0 ⟹ C_{η}^{•−2} ⊗_{A_{η}} I ⟹ E_{η}^{•} ⊗_{A_{η}} I ⟹ E^{•} ⊗_{A_{η}} I ⟹ 0 (3.11)

which induces a long exact sequence of cohomologies

(3.12) 0 ⟶ h^{1}(E_{η}^{•} ⊗_{A_{η}} I) ⟶ h^{1}(E^{•} ⊗_{A} I) ⟶ I ⊗_{A_{η}} O_{S_{η}}(L_{η})

⟶ h^{2}(E_{η}^{•} ⊗_{A_{η}} I) ⟶ h^{2}(E^{•} ⊗_{A} I) ⟶ 0.

In particular, h^{k}(E_{η}^{•} ⊗_{A_{η}} I) = 0 except k = 1, 2.

Proposition 3.7. First, the functor of the first order deformations of S_{η} is naturally isomorphic to the functor 𝔥^{1}(E_{η}^{•}). Secondly, there is a natural obstruction theory to deformation of the families of predeformable morphisms f|_{S_{η}} : X |_{S_{η}} → W [n] ×_{A^{n+1}} H_{l} taking values in 𝔥^{2}(E_{η}^{•}). Finally, such obstruction theory is perfect.

Proof. We will omit the proof here because it is parallel to the treatment of the obstruction theory of 𝔐(𝔚_{0}, Γ). q.e.d.

Proposition 3.8. The perfect obstruction theory constructed in Proposition 3.7 induces a perfect obstruction theory of 𝔐(𝔚_{0}, η), which in turn defines a natural virtual moduli cycle [𝔐(𝔚_{0}, η)]^{virt}.

Proof. The proof is parallel to the construction of the perfect obstruction theory of 𝔐(𝔚, Γ) and of the virtual moduli cycle [𝔐(𝔚, Γ)]^{virt}. We shall omit the details here. q.e.d.

We next work out the obstruction theory of 𝔐(𝔜_{1}^{rel} ⊔ 𝔜_{2}^{rel}, η). We let S = Spec A → 𝔐(W [n], Γ)^{st} be an η-admissible chart and let f : X → W [n] be the universal family over S. We then pick a sufficiently fine covering (U_{α}/V_{α}, f_{α}) indexed by Λ. Following Section 1.2, we pick the complex [F^{0} ⟶^{d} F^{1}] as in (1.5) and pick extensions ζ_{α} as defined before (1.18). Based on these data we can form the complex E^{•} as in (1.19) so that its cohomology is part of the obstruction theory of S. We let S_{0} = Spec A_{0} = S ×_{𝔐(𝔚,Γ)} 𝔐(𝔜_{1}^{rel} ⊔ 𝔜_{2}^{rel}, η) and let f_{0} :X_{0} → W [n] be the restriction of f to fibers over S_{0}. We let l ∈ [n + 1] be the integer associated to η defined in Definition 3.3. By definition, the family f_{0} can be decomposed into two families of relative stable morphisms of types Γ_{1} and Γ_{2} respectively along a multi-section Σ ⊂ X_{0,node} over S_{0}. We let U_{0,α} = U_{α} ×_{S} S_{0}, V_{0,α} = V_{α} ×_{S} S_{0} and let f_{0,α} = f_{α}|_{U_{0,α}}. Note that Σ is étale over S_{0}. Now let X̂_{0} be the formal completion of X_{0} along Σ ⊂ X_{0}.

Then since Σ → S is finite and étale, and since Σ ⊂ X_{0} is a multiple section of the nodal points of the fibers of X_{0}/S_{0}, the extension sheaf Ext _{X_{0}}^{1}(Ω_{X̂_{0}/S_{0}}, OX̂_{0}) is an invertible sheaf of O_{Σ}-modules. We denote this sheaf by M_{Σ}. Then we have a natural homomorphism

Ext_{X_{0}}^{1}(Ω_{X_{0}/S_{0}}, O_{X_{0}}) ⟶ ρ_{∗}(Ext _{X_{0}}^{1}(Ω_{X_{0}/S_{0}}, O_{X_{0}})) ⟶^{proj} ρ_{∗}M_{Σ},

where ρ: X_{0} → S_{0} and \overline{ρ}: Σ → S_{0} are the projections, which defines a canonical homomorphism

F^{1} ⊗_{A} A_{0} ⟶ Ext_{X_{0}}^{1}(Ω_{X_{0}/S_{0}}, O_{X_{0}}) ⟶ \overline{ρ}_{∗}M_{Σ}. (3.13)

Clearly, the composite of (3.13) is surjective. We let F_{η,0}^{0} = F^{0} ⊗_{A} A_{0} and let F_{η,0}^{1} be the kernel of (3.13). The module F_{η,0}^{1} is a free A_{0}-module and F^{0} ⊗_{A} A_{0} → F^{1} ⊗_{A} A_{0} factor through F_{η,0}^{0} → F_{η,0}^{1}.

We now construct the complex that will give the obstruction theory of S_{0} (which is a chart of 𝔐(𝔜_{1}^{rel} ⊔ 𝔜_{2}^{rel}, η)). We let E^{•} be the complex associated to the family f and the covering Λ mentioned before. We let E_{η,0}^{k} ≜ E^{k} ⊗_{A} A_{0} for k ≠ 1 and let E_{η,0}^{1} be the kernel of the composite E^{1} ⊗_{A} A_{0} ⟶^{pr} F^{1} ⊗_{A} A_{0} → \overline{ρ}_{∗}M_{Σ}. Clearly, the differentials in E^{•} induce differentials in E_{η,0}^{•}.

Proposition 3.9. Lemma 1.15 holds true for the complex E_{η,0}^{•}. Proposition 3.7 holds true for the charts S_{0} with E_{η}^{•} replaced by E_{η,0}^{•}. Proposition 3.8 holds true for the moduli stack 𝔐(𝔜_{1}^{rel} ⊔ 𝔜_{2}^{rel}, η).

Proof. We shall sketch the construction of the obstruction classes. The remainder part of the proof is similar to that of Propositions 3.7 and 3.8 and will be omitted. Here we will follow closely the convention introduced in the proof of Proposition 1.18. Let ξ = (B, I, φ) be an object in 𝔗𝔯𝔦_{S_{0}}. Let f_{T} : X_{T} → W [n] be the pullback of f via T → S_{0}, where T = Spec B/I, and let Σ_{T} = X_{T} ×_{X_{0}} Σ. By the definition of the subscheme S_{0} ⊂ S, the formal completion of X_{T} along Σ_{T} is isomorphic to Spec 𝕜[[z_{1}, z_{2}]]/(z_{1}z_{2})×Σ_{T}, at least after shrinking S if necessary. Now let T̃ = Spec B. Then by the deformation theory of nodal curves, we can find a flat extension XT̃/T̃ of X_{T} /T so that the formal completion of XT̃ along Σ_{T} is isomorphic to Spec 𝕜[[z_{1}, z_{2}]]/(z_{1}z_{2}) × ΣT̃, where ΣT̃ is étale over T̃ so that ΣT̃ ×T̃ T ≡ Σ_{T}. In other words, the multiple section Σ_{T} ⊂ X extends to a multiple section ΣT̃ ⊂ XT̃ and the extended family XT̃ can be decomposed along ΣT̃. Once we have chosen such an extension, to each α ∈ Λ we can pick a predeformable extension h̃_{α}: Ũ_{0,α} → W [n] of f_{T,α} so that the composite Ũ_{0,α} → W [n] → A^{n+1} factor through the l-th coordinate hyperplane H_{l} ⊂ A^{n+1}. We let b_{αβ} be as defined in (1.21). Then b = {b_{αβ}} ∈ E^{2} ⊗_{A} I ≡ E_{η,0}^{2} ⊗_{A_{0}} I. We define

𝔬𝔟(ξ) = [b] ∈ h^{2}(E_{η,0}^{•} ⊗_{A_{0}} I).

It is direct to check that this defines an obstruction class. It is routine to check that such choices of obstruction classes satisfies the base change property. q.e.d.

Applying the basic construction of virtual moduli cycles formulated in the previous section, we obtain cycles [𝔐(𝔚_{0}, η)]^{virt} and [𝔐(𝔜_{1}^{rel} ⊔ 𝔜_{2}^{rel}, η)]^{virt}.

We are now ready to state the degeneration formula of the Gromov-Witten invariants of the family W. Let η = (Γ_{1}, Γ_{2}, I) be an admissible triple in Ω. Let μ_{i} be the weight of the i-th root of Γ_{1} and Γ_{2}, which are the same. We define the multiplicity of η to be m(η) = ∏_{i=1}^{r} μ_{i}.

Lemma 3.10. We have the identity

c_{1}(L_{0}, s_{0})[𝔐(𝔚, Γ)]^{virt} = [𝔐(𝔚_{0}, Γ)]^{virt} ∈ A_{∗}𝔐(𝔚_{0}, Γ).

Lemma 3.11. We have the identity

[𝔐(𝔚_{0}, η)]^{virt} = c_{1}(L_{η}, s_{η})[𝔐(𝔚, Γ)]^{virt} ∈ A_{∗}𝔐(𝔚_{0}, η).

Lemma 3.12. Under the natural isomorphism

A_{∗}𝔐(𝔜_{1}^{rel} ⊔ 𝔜_{2}^{rel}, η) ≅ A_{∗}𝔐(𝔚_{0}, η)

induced by the homeomorphism 𝔐(𝔜_{1}^{rel} ⊔ 𝔜_{2}^{rel}, η) ≃ 𝔐(𝔚_{0}, η),

m(η)[𝔐(𝔜_{1}^{rel} ⊔ 𝔜_{2}^{rel}, η)]^{virt} = [𝔐(𝔚_{0}, η)]^{virt} ∈ A_{∗}𝔐(𝔚_{0}, η).

Using the identity of C-divisors (L_{0}, s_{0}) = ⊗_{η∈Ω/∼}(L_{η}, s_{η}), we have

Corollary 3.13.

[𝔐(𝔚_{0}, Γ)]^{virt} = ∑_{η∈Ω/∼} m(η)[𝔐(𝔜_{1}^{rel} ⊔ 𝔜_{2}^{rel}, η)]^{virt}.

We now state how the virtual moduli cycle [𝔐(𝔜_{1}^{rel} ⊔ 𝔜_{2}^{rel}, η)]^{virt} is related to [𝔐(𝔜_{i}^{rel}, Γ_{i})]^{virt}. Using the natural evaluation morphism q_{i}:𝔐(𝔜_{i}^{rel}, Γ_{i}) → D^{r} we form the Cartesian diagram

𝔐(𝔜_{1}^{rel}, Γ_{1}) ×_{D^{r}} 𝔐(𝔜_{2}^{rel}, Γ_{2}) ⟶ 𝔐(𝔜_{1}^{rel}, Γ_{1}) × 𝔐(𝔜_{2}^{rel}, Γ_{2})
↓  ↓
D^{r}  ⟶^{Δ}  D^{r} × D^{r}

(3.14)

Here the arrow Δ is the diagonal morphism. Let Φ_{η} be the finite étale morphism in (3.2), which has pure degree | Eq(η)| (see [23, Section 4]).

Lemma 3.14. We have the identity

\frac{1}{| Eq(η)|}Φ_{η∗}Δ^{!}([𝔐(𝔜_{1}^{rel}, Γ_{1})]^{virt} × [𝔐(𝔜_{2}^{rel}, Γ_{2})]^{virt})

= [𝔐(𝔜_{1}^{rel} ⊔ 𝔜_{2}^{rel}, η)]^{virt}.

We will prove these lemmas in the next section.

The main degeneration formula of the Gromov-Witten invariants of W follows immediately from these lemmas and the first version of the degeneration formula proved in the previous subsection.

Theorem 3.15. Let the notation be as before. Then as elements in A_{∗}𝔐(𝔚_{0}, Γ),

[𝔐(𝔚_{0}, Γ)]^{virt}

= ∑_{η∈Ω/∼} \frac{m(η)}{| Eq(η)|} Φ_{η∗}Δ^{!}([𝔐(𝔜_{1}^{rel}, Γ_{1})]^{virt} × [𝔐(𝔜_{2}^{rel}, Γ_{2})]^{virt}).

Finally, we state the numerical corollary of this theorem. Let ȷ_{i} : Y_{i} → W be the inclusion and let

ȷ_{i}^{∗}:H_{C}^{0}(R^{∗}π_{∗}ℚ_{W})^{×k} → H^{∗}(Y_{i}, ℚ)^{×k}

be the induced pullback homomorphism. Now let η = (Γ_{1}, Γ_{2}, I) ∈ Ω be any admissible triple. For i = 1 or 2, we let 𝔐_{Γ_{i}^{∘}} be the moduli space of stable curves of topological type Γ_{i}^{∘} (See the definition before (2.9)). It is naturally a Deligne-Mumford stack. Further, we have a natural local immersion of stacks

ϕ_{η} : 𝔐_{Γ_{1}^{o}} × 𝔐_{Γ_{2}^{o}} ⟶ 𝔐_{g,k}

that associates to any pair of curves (C_{1}, C_{2}) ∈ 𝔐_{Γ_{1}^{o}} × 𝔐_{Γ_{2}^{o}} the gluing C_{1}⊔C_{2} by identifying the i-th distinguished marked point of C_{1} with the i-th distinguished marked point of C_{2} for all i. Now let β ∈ H^{∗}(𝔐_{g,k}). We assume β has the following Kunneth type decomposition

ϕ_{η}^{∗}(β) = ∑_{j∈K_{η}} β_{η,1,j} ⊠ β_{η,2,j}, β_{i,j} ∈ H^{∗}(𝔐_{Γ_{i}^{o}}).

Corollary 3.16. Let W/C be the family and let Γ = (g, b, k) be as before. Then for any closed point ξ ≠ 0 ∈ C, α ∈ H_{C}^{0}(R^{∗}π_{∗}ℚ_{W})^{×k} and β ∈ H^{∗}(𝔐_{g,n}) as before,

Ψ_{Γ}^{W_{ξ}}(α(ξ), β)

= ∑_{η∈Ω/∼} \frac{m(η)}{| Eq(η)|} ∑_{j∈K_{η}} [Ψ_{Γ_{1}}^{Y_{1}^{rel}}(ȷ_{1}^{∗}α, β_{η,1,j}) • Ψ_{Γ_{2}}^{Y_{2}^{rel}}(ȷ_{2}^{∗}α, β_{η,2,j})]_{0} .

Here • is the intersection of the homology groups

H_{∗}(D^{r}) × H_{∗}(D^{r}) ⟶^{∩} H_{∗}(D^{r})

and [γ]_{0} is the degree of the degree 0 part of the homology class γ ∈ H_{∗}(D^{r}).

4. Proof of the main theorem

The goal of this section is to prove Lemma 3.12-3.14. In essence, the proofs of these lemmas (except Lemma 3.12) rely on the comparison of the virtual moduli cycles of stacks with the virtual moduli cycles of their substacks. This is precisely the situation studied in [26, Lemma 3.4], except that there we used the existence of certain locally free sheaves to construct the virtual moduli cycles. To prove Lemma 3.12, we need to study the situation more general than the one studied. In the first subsection we will revise [26, Lemma 3.4] to cover all the situations we need.

4.1 Comparison of the virtual moduli cycles

Let M → N be a representable morphism of stacks. In this subsection, we assume M is a DM-stack having a perfect-obstruction theory with the associated obstruction sheaves Ob_{M}. For N we need to consider two possibilities: One is when N is a DM-stack having a perfect-obstruction theory with the obstruction sheaf Ob_{N}. The other is when N is a closed substack of a smooth Artin stack Q defined by the vanishing of a section s of a vector bundle F on Q.

Let S = {S_{α}}_{Λ} be an atlas of N and let {(F_{α}^{•}, 𝔬𝔟_{S_{α}})}_{Λ} be the data associated to the perfect-obstruction theory of N. In case N is a DMstack this is specified in the Definition 2.1. In case N is an Artin stack, we assume that there is an atlas \overline{S} = \{\overline{S}_{α}\}_{Λ} of Q so that S_{α} = \overline{S}_{α} ×_{Q} N. Then we simply take F_{α}^{•} = [F_{α}^{1} → F_{α}^{2}] to be F_{α}^{1} = O_{S_{α}}(T \overline{S}_{α}), F_{α}^{2} = O_{S_{α}}(F |_{S_{α}}) and the arrow F_{α}^{1} → F_{α}^{2} to be the one induced by the differential of the section s ∈ H^{0}(F). The obstruction assignment 𝔬𝔟_{S_{α}} taking values in Ob_{S_{α}} = h^{2}(F_{α}^{•}) is the obvious one induced by the defining equation s. Since M → N is representable, S_{α} ×_{N} M is a scheme. For each α ∈ Λ we pick an affine étale universal open R_{α} → S_{α} ×_{N} M. Without loss of generality, we can assume {R_{α}}_{Λ} is a covering of M in the sense that the image R_{α} ≜ ρ_{α}(R_{α}) of the tautological p_{α}:R_{α} → M is an open substack and the collection {R_{α}}_{Λ} forms an open covering of M. Note that when N is a DM-stack, R_{α} → M are étale and then {R_{α}}_{Λ} forms an étale cover of M. In case N is an Artin stack, then R_{α} → M are smooth morphisms. In this case we shall view {R_{α}}_{Λ} as an atlas in the smooth site^{16} of M.

We let {(E_{α}^{•}, 𝔬𝔟_{R_{α}})}_{Λ} be the date given by the perfect obstruction theory of M associated to the covering {R_{α}}_{Λ}. Namely, each (E_{α}^{•}, 𝔬𝔟_{R_{α}}) is a perfect obstruction theory of R_{α}, the sheaves Coker{E_{α}^{1} → E_{α}^{2}} descends to the sheaf Ob_{M} of O_{M}-modules and the obstruction assignments {𝔬𝔟_{R_{α}}} are compatible over all R_{αβ} = R_{α} ×_{M} R_{β}.

We next assume R_{α} → S_{α} admits a perfect relative obstruction theory given by (L_{α}^{•}, 𝔬𝔟_{R_{α}/S_{α}}) as defined in Definition 1.11. We say M → N admits a perfect relative obstruction theory if we can choose {(L_{α}^{•}, 𝔬𝔟_{R_{α}/S_{α}})}_{Λ} so that the relative obstruction sheaves Ob_{R_{α}/S_{α}} ≜ Coker{L_{α}^{1} → L_{α}^{2}} descends to a global sheaf on M and the obstruction assignments 𝔬𝔟_{R_{α}/S_{α}} are compatible on the overlaps R_{αβ}.

Definition 4.1. The perfect (relative) obstruction theories {E_{α}^{•}, 𝔬𝔟_{R_{α}}}, {F_{α}^{•}, 𝔬𝔟_{S_{α}}} and {L_{α}^{•}, 𝔬𝔟_{R_{α}/S_{α}}} are said to be compatible if to each α ∈ Λ there is an exact triangle of complexes

⟹ L_{α}^{•} ⟹ E_{α}^{•} ⟹ F_{α}^{•} ⊗_{O_{Sα}} O_{R_{α}} ⟹ L_{α}^{•+1} ⟹ (4.1) which induces a long exact sequence of cohomologies

16Namely, the open covering are univeral open smooth morphisms.

→ h^{i}(L_{α}^{•} ⊗ I) ⟶^{τ_{1,i}} h^{i}(E_{α}^{•} ⊗ I) ⟶^{τ_{2,i}} h^{i}(F_{α}^{•} ⊗_{O_{Sα}} I) ⟶^{τ_{0,i+1}} h^{i+1}(L_{α}^{•} ⊗ I) → (4.2)

for any sheaf of O_{R_{α}}-modules I that satisfies the following properties:

(1) The first part of the above exact sequence is identical to the exact sequence

0 ⟶ 𝔇𝔢𝔣_{R_{α}/S_{α}}^{1}(I) ⟶ 𝔇𝔢𝔣_{R_{α}}^{1}(I) ⟶ 𝔇𝔢𝔣_{S_{α}}^{1}(I) ⟶

under the canonical isomorphisms 𝔇𝔢𝔣_{·}^{1}(I) = h^{1}(· ⊗ I) given by the definition of the perfect obstruction theories.

(2) Let ξ be any object in 𝔗𝔯𝔦_{R_{α}}, which is also an object in 𝔗𝔯𝔦_{S_{α}}. Then under the arrow τ_{2,2} in (4.2) we have 𝔬𝔟_{S_{α}}(ξ) = τ_{2,2}(𝔬𝔟_{R_{α}}(ξ)).

(3) Let ξ = (B, I, φ_{0}) be an object in 𝔗𝔯𝔦_{R_{α}/S_{α}}, which is also an object in 𝔗𝔯𝔦_{R_{α}}. Then 𝔬𝔟_{R_{α}}(ξ) = τ_{2,1}(𝔬𝔟_{R_{α}/S_{α}}(ξ)). Further, suppose 𝔬𝔟_{R_{α}}(ξ) = 0 and φ_{0} : Spec B/I → R_{α} extends to φ : Spec B → R_{α}. Let e ∈ h^{1}(F_{α}^{•} ⊗_{B/I} I) be the difference of the tautological Spec B → Spec S_{α} and the composite of Spec B ⟶^{φ} Spec R_{α} and Spec R_{α} ⟶^{pr} Spec S_{α}. Then 𝔬𝔟_{R_{α}/S_{α}}(ξ) = τ_{0,2}(e).

Finally, the collection of exact sequences h^{2}(L_{α}^{•}) → h^{2}(E_{α}^{•}) → h^{2}(F_{α}^{•}) ⟶ 0 descends to an exact sequence of sheaves

Ob_{M/N} ⟶ Ob_{M} ⟶ Ob_{N} ⊗_{O_{N}} O_{M} ⟶ 0.

The goal of this subsection is to show that with the data given in Definition 4.1, we can construct a class [M, N]^{virt} ∈ A_{∗}M, called the relative virtual moduli cycle. We will then show that it is equal to [M]^{virt} in A_{∗}M. This allows us to give a different interpretation of [M]^{virt}, useful in the proof of the key lemmas in the previous section.

We first study the local situation. Let R_{α} → M and S_{α} → N be respective charts as described before. Let p ∈ M be any closed point, q ∈ N be the image of p, \overline{p} ∈ R_{α} be a lift of p ∈ M and \overline{q} ∈ S_{α} be the image of \overline{p}. We let T_{1} = h^{1}(E_{α}^{•} ⊗ 𝕜\overline{p}), T_{2} = h^{1}(F_{α}^{•} ⊗ 𝕜\overline{q}) and let T_{1/2} = h^{1}(L_{α}^{•}⊗𝕜\overline{p}). Similarly, we let O_{1} = h^{2}(E_{α}^{•}⊗𝕜\overline{p}), O_{2} = h^{2}(F_{α}^{•}⊗𝕜\overline{q}) and let O_{1/2} = h^{2}(L_{α}^{•} ⊗ 𝕜\overline{p}), all implicitly depending on the lift \overline{p}. Note that they fit into the exact sequence

0 ⟶ T_{1/2} ⟶ T_{1} ⟶ T_{2} ⟶^{δ} O_{1/2} ⟶ O_{1} ⟶ O_{2} ⟶ 0 (4.3)

induced by (4.1). We now let T = T_{1} ⊕ T_{2}/ ker(δ) and pick a surjective homomorphism T → T_{2} that is an extension of T_{1} → T_{2}. Next we let O = O_{1} ⊕Im(δ). We then pick an injective homomorphism η:O_{1/2} → O so that the composite O_{1/2} → O → O_{1/2} is the identity map. This way we have two exact sequences

0 → T_{1/2} → T → T_{2} → 0 and 0 → O_{1/2} → O → O_{2} → 0. (4.4)

Further, with various homomorphisms chosen and the isomorphism T_{2}/ ker(δ) ≅ Im(δ), we have the following induced exact sequence

0 ⟶ T_{1} ⟶ T ⟶ O ⟶ O_{1} ⟶ 0.

In the following for any vector space V we denote by 𝕜[[V]] the ring of formal power series lim_{n} ⊕_{i=0}^{n}S^{i}(V). We now let

f ∈ 𝕜[[T^{∨}]] ⊗ O, g ∈ 𝕜[[T_{2}^{∨}]] ⊗ O_{2} and h ∈ 𝕜[[T^{∨}]]/(g) ⊗ O_{1/2} (4.5)

be the Kuranishi maps of the (relative) obstruction theories of R_{α}, of S_{α} and of R_{α}/S_{α} at \overline{p} (or \overline{q}) respectively (see [26, Lemma 3.10]). By abuse of notation, we denote by (g) the idea generated by the components of g in 𝕜[[T_{2}^{∨}]] and in 𝕜[[T^{∨}]] via the inclusion 𝕜[[T_{2}^{∨}]] → 𝕜[[T^{∨}]].

Lemma 4.2. We can choose the Kuranishi maps f, g and h so that:

(1) φ_{1}(f_{p}) = τ(g_{p}) under the naturally induced maps φ_{1} :𝕜[[T^{∨}]]⊗O → 𝕜[[T^{∨}]] ⊗ O_{2} and τ:𝕜[[T_{2}^{∨}]] ⊗ O_{2} → 𝕜[[T^{∨}]] ⊗ O_{2}.

(2) The differential dh_{p}(0) : T → O_{1/2} is identical to the composite T → T_{2}/ ker(δ) → O_{1/2} induced by δ in (4.3).

(3) Let h̃ and f̃ in 𝕜[[T_{1/2}^{∨}]]/(g)⊗O be the images of h and f under the

obvious maps induced by the arrows mentioned before, then f̃ = h̃.

Proof. The proof follows from the construction of Kuranishi maps, as was demonstrated in [26, Lemma 3.10]. q.e.d.

We now let X = Spec 𝕜[[T^{∨}]], V_{1/2} = O_{1/2} × X, V = O × X and let V_{2} = O_{2} × X, all viewed as vector bundles (or their total spaces) over X. We let ρ_{1} : V_{1/2} → V and ρ_{2} : V → V_{2} be the vector bundle homomorphisms induced by arrows in (4.4). Then V_{2} is the quotient vector bundle of V by V_{1/2}. We consider the subscheme Γ ⊂ V_{1} that is the graph of f, consider Y = V_{1/2} ×_{X} Γ ⊂ V_{1/2} ×_{X} V and consider

Θ_{1} = Ṽ_{1/2} ×_{V_{1/2}×_{X}V} Y ⊂ Y, Θ_{2} = Ṽ_{1} ×_{V_{1/2}×_{X}V} Y ⊂ Y.

Here Ṽ_{1/2} is the image scheme of the immersion 1_{V_{1/2}} × ρ_{1} : V_{1/2} → V_{1/2} ×_{X} V and Ṽ ≜ 0_{V_{1/2}} × V ⊂ V_{1/2} ×_{X} V, where 0_{V_{1/2}} is the zero section of V_{1/2}. Note that Ṽ_{1/2} and Ṽ are isomorphic to V_{1/2} and V, respectively. Following [26, Page 145], we denote the normal cone to C_{Θ_{2}/Y} ×_{Y} Θ_{1} in C_{Θ_{2}/Y} ^{17} by B(p)_{1} and denote the normal cone to C_{Θ_{1}/Y} ×_{Y} Θ_{2} in C_{Θ_{1}/Y} by B(p)_{2}. Both B(p)_{1} and B(p)_{2} are subcones in V̂_{1/2} ×X̂ V̂, where X̂ = Spec 𝕜[[T_{1}^{∨}]]/(f) and V̂_{i} = V_{i} ×_{X} X̂. As argued in [26, 146], based on the work of [38] (see also the recent [21]) there is a canonical rational equivalence^{18} Q(p) ∈ W_{∗}(V̂_{1/2} ×X̂ V̂) so that

∂_{0}Q(p) = B(p)_{1} and ∂_{∞}Q(p) = B(p)_{2}. (4.6)

The cones B(p)_{1} and B(p)_{2} have the following interpretations as shown in [26, page 145]. Let D(p)_{1} ⊂ V̂ be the normal cone to X̂ in X, then

B(p)_{1} = ϕ_{1}^{∗}D(p)_{1} (4.7)

where ϕ_{1} : V̂_{1/2} ×X̂ V̂ ⟶ V̂ is the projection. Next, we let W = Spec 𝕜[[T_{2}^{∨}]] and let Ŵ = Spec 𝕜[[T_{2}^{∨}]]/(g). We then form the normal cone to C_{Ŵ /W} ×_{W} X̂ in C_{Ŵ /W} ×_{W} X, denoted by D(p)_{2}. It is naturally a subcone in V̂_{1/2} ×X̂ V̂_{2}. Then

B(p)_{2} = ϕ_{2}^{∗}D(p)_{2}, (4.8)

where ϕ_{2} = (1, ρ_{2}) : V̂_{1/2} ×X̂ V̂ ⟶ V̂_{1/2} ×X̂ V̂_{2}.

We caution that all the objects so far constructed depends on the lift \overline{p}, on the choices of arrows before (4.4) and on the Kuranishi maps. Later we will show that they are canonical in certain degree, up to the symmetry Aut(p).

17For closed subscheme A ⊂ B, C_{A/B} is the normal cone to A in B.

18In this paper we use the convention that a rational equivalence Q ∈ W_{∗}Z is a cycle in Z_{∗}(Z × P^{1}) so that all its irreducible components are flat over P^{1}. We then define ∂_{0}Q and ∂_{∞}Q to be Q ∩ Z × {0} and Q ∩ Z × {∞} respectively. In case φ: Z → W is a flat morphism, we denote by φ^{∗}Q the flat pullback of the rational equivalence.

In the following, we first construct the relative moduli cycle [M, N]^{virt} and several related cycles based on the collection of the infinitesimal models D(p)_{2}, etc. First at each p ∈ M the cone

D(p)_{2} ×X̂ 0 ⊂ (V̂_{1/2} ×X̂ V̂_{2}) ×_{X} 0

is independent of the choices of the Kuranishi maps f, g and h and is invariant under the natural Aut(p) action. (Here X̂ is the subscheme defined before that depends on p implicitly.) Similar statements hold for the cycles D(p)_{1} ∈ Z_{∗}V̂, the cycles B(p)_{i} ∈ Z_{∗}(V̂_{1/2} ×X̂ V̂) and the rational equivalence Q(p) ∈ W_{∗}(V̂_{1/2} ×X̂ V̂). These were proved in [26, Section 3]. Secondly, for each α we pick a pair of locally free subsheaves^{19} L_{α} ⊂ E_{α} over R_{α}, viewed as a complex [L_{α} → E_{α}], and a surjective homomorphism of complexes

[L_{α} → E_{α}] ⟹ [Ob_{R_{α}/S_{α}} → Ob_{R_{α}}]. (4.9)

For simplicity, we denote by W_{α} the vector bundle Vect(L_{α} ⊕ E_{α}). As shown in [26, Section 3] there is a unique cycle B_{i,α} ∈ Z_{∗}W_{α} so that the collection B_{i} = {(R_{α}, B_{i,α}, L_{α} ⊕ E_{α})}_{Λ} satisfies the cycle consistency criteria for the infinitesimal models B(p)_{i} over the pair (M, Ob_{M/N} ⊕ Ob_{M}). We let [B_{1}] and [B_{2}] be the associated class in A_{∗}M following the basic construction. Next we let F_{α} ≜ E_{α}/L_{α} be the quotient sheaf, which is locally free by our choice. Then Ob_{S_{α}} ⊗_{O_{Sα}} O_{R_{α}} is canonically a quotient sheaf of F_{α}. Again following [26, Section 3] we can find a unique cone cycle D_{2,α} ⊂ Vect(L_{α} ⊕ F_{α}) so that the collection D_{2} = {(R_{α}, D_{2,α}, L_{α} ⊕ F_{α})} satisfies the cycle consistency criteria with the infinitesimal models D(p)_{2} ⊂ V̂_{1/2} ×X̂ V̂_{2} over the pair (M, Ob_{M/N} ⊕ ρ^{∗}Ob_{N}), where ρ:M → N is the projection. Thus by applying the basic construction to this collection we obtain the class [D_{2}] ∈ A_{∗}M. We will call the class [D_{2}] the relative cycle and denoted it by [M, N]^{virt}.

Lemma 4.3. We have [M]^{virt} = [M, N]^{virt} in A_{∗}M.

To prove the lemma we need to construct a rational equivalence [Q] ∈ W_{∗}M (or equivalently a class [Q] ∈ A_{∗}M × P^{1}) so that ∂[Q] provides the identity in the lemma. Here when [Q] is a class, we define ∂[Q] = ∂_{0}[Q] − ∂_{∞}[Q] with ∂_{0}[Q] the image of the Gysin map 0^{!}[Q] associated to 0 → P^{1}. The ∂_{∞} is defined similarly. The proof is parallel to that of [26, Lemma 3.4] and will occupy the rest of this subsection.

19In this paper by a pair of locally free sheaves we mean a locally free subsheaf of a locally free sheaf with locally free quotient sheaf, all of finite ranks.

We need to provide a revised proof since the current construction does not rely on the existence of a global vector bundle as was assumed in [26].

We first set up the notation relevant to the construction of the cycles [B_{i}], etc, following the basic construction of the virtual cycles. First, in constructing the class [B_{i}] we first built the index set Ξ(B_{i}). Each a ∈ Ξ(B_{i}) associates to an integral closed substack Y_{a} ⊂ M, a multiplicity m_{a}, a maximal open substack ȷ_{a}:Y_{a}^{0} → Y_{a} so that ȷ_{a}^{∗}Ob_{M/N}⊕ȷ_{a}^{∗}Ob_{M} is locally free. Over Y_{a}^{0} we have a canonical cone representative N_{a}^{0} ⊂ F_{a}^{0}, where F_{a}^{0} is the vector bundle (stack) Vect(ȷ_{a}^{∗}Ob_{M/N} ⊕ ȷ^{∗}Ob_{M}) over Y_{a}^{0}. We then pick a projective variety φ_{a} : Y_{a} → Y_{a} generically finite over Y_{a}, a locally free sheaf F_{a} of O_{Y_{a}}-modules and a quotient sheaf homomorphism F_{a} → φ_{a}^{∗}(Ob_{M/N} ⊕ Ob_{M}). We form the vector bundle F_{a} = Vect(F_{a}) with the induced flat morphism F_{a}|_{φ_{a}^{−1}(Y_{a}^{0})} → F_{a}^{0}. We let N_{a} ⊂ F_{a} be the closure of the flat pullback of N_{a}^{0}. Then [B_{i}] is the sum of m_{a}ξ(a) over all a ∈ Ξ(B_{i}), where ξ(a) is deg(φ_{a})^{−1}φ_{a∗}0_{F_{a}}^{!}[N_{a}].

We set up the notation for the cycle [D_{i}] according to the same rule. We first prove [B_{1}] = [M]^{virt}. This identity follows from the relation (4.7). The actual proof goes as follows: First the class [D_{1}] = [M]^{virt} (see [26, Section 3]). From (4.7) the index sets Ξ(B_{1}) ≡ Ξ(D_{1}) naturally. For a ∈ Ξ(B_{1}) with \overline{a} ∈ Ξ(D_{1}) the corresponding element, their base substacks Y_{a} = Y\overline{a}. Further the intrinsic representative N_{a}^{0} is the flat pullback of N0 ; \overline{a} under an obvious homomorphism of vector bundles. Based on these, we can choose Y_{a} = Y\overline{a}, choose locally free sheaves F_{a} = F\overline{a} and then the cone representatives N_{a} ≡ N\overline{a}. This proves ξ(a) = ξ(\overline{a}) and hence [B_{1}] = [D_{1}] = [M]^{virt}. Since the proof is straightforward, we will omit the details here. For the same reason, we prove [B_{2}] = [M, N]^{virt} based on the identity (4.8). Again we will omit the details here.

We now construct the required cycle [Q] so that

∂_{0}[Q] = [B_{1}] and ∂_{∞}[Q] = [B_{2}]. (4.10)

For each α ∈ Λ we fix the pair L_{α} ⊂ E_{α} over R_{α} as in (4.9). As before we let W_{α} = Vect(L_{α} ⊕ E_{α}). We claim that we can find a collection of rational equivalence Q_{α} ⊂ W_{∗}W_{α} indexed by α ∈ Λ that satisfy the following existence lemma.

Lemma 4.4. Let α ∈ Λ be any element. Then there is a unique rational equivalence Q_{α} ∈ W_{∗}W_{α} of which the following holds. Let p ∈ M be any point and let \overline{p} be a lift of p in some chart R_{β}. We let V̂_{1/2}, V̂ and V̂_{2} be the vector bundles over X̂ (associated to \overline{p}) defined before and after Lemma 4.2. Let R_{α} be the chart, let R_{α,p} = R_{α} ×_{M} p and let R̂_{α} be the formal completion of R_{α} along R_{α,p}. We also pick a morphism R̂_{α} → X̂ that commutes with R̂_{α} → M and X̂ → M. Now let O_{1/2} ↪ O be the pair in (4.4) associated to \overline{p}. Then up to Aut(p) there are canonical induced homomorphisms of vector bundles over R_{α,p}

O_{1/2} × R_{α,p} → h^{2}(L_{α})|_{R_{α,p}}, O × R_{α,p} → h^{2}(E_{α})|_{R_{α,p}} and
L_{α}|_{R_{α,p}} → O_{1/2} × R_{α,p}.

We then pick a surjective homomorphism φ_{2} : E_{α}|_{R_{α,p}} → O × R_{α,p} so that the following diagram of the complexes of vector bundles over R_{α,p}

\xymatrix{[L_{α}|_{R_{α,p}} → E_{α}|_{R_{α,p}}] \ar@{=>}[r] \ar@{=>}[dr] & [O_{1/2} × R_{α,p} → O × R_{α,p}] \ar@{=>}[d] \\ & [h^{2}(L_{α}^{•})|_{R_{α,p}} → h^{2}(E_{α}^{•})|_{R_{α,p}}]} (4.11)

is commutative. Then there is a vector bundle homomorphism

Φ_{1} : W_{α} ×_{R_{α}} R̂_{α} ⟶ (V̂_{1/2} ×X̂ V̂) ×X̂ R̂_{α}

(recall W_{α} = Vect(L_{α} ⊕ E_{α})) extending the obvious homomorphism

W_{α} ×_{R_{α}} R_{α,p} ⟶ (V_{1/2} × V) × R_{α,p}

induced by the φ_{2} before (4.11) so that

Φ_{1}^{∗}(ψ^{∗}Q(p)) = Φ_{2}^{∗}(Q_{α}),

where ψ:(V̂_{1/2} ×X̂ V̂) ×X̂ R̂_{α} → V̂_{1/2} ×X̂ V̂ and Φ_{2} :W_{α} ×_{R_{α}} R̂_{α} → W_{α} are the obvious (flat) projections.

We will call this lemma the rational equivalence consistency criteria. This lemma is proved in [26, Section 3] for the case where M is a

quotient stack. The general case is exactly the same. We will not repeat the argument here.

We now fix the collection Q = {(R_{α}, Q_{α}, L_{α} ⊕ E_{α})}_{Λ}. To each α ∈ Λ we let Ξ(Q_{α}) be the set of irreducible components of Q_{α}. For each a ∈ Ξ(Q_{α}), we let T_{a} ⊂ Q_{α} be the corresponding irreducible component, let m_{a} be the multiplicity of Q_{α} along T_{a} and let Y_{a} ⊂ M be the base substack of a, which is the closure of the image of T_{a} → M. We then let ι_{a} : Y_{a}^{0} → Y_{a} be an open substack so that ι_{a}^{∗}Ob_{M/N}, ι_{a}^{∗}Ob_{M} and K_{a}^{0} = ker{ι_{a}^{∗}Ob_{M/N} → ι_{a}^{∗}Ob_{M}} are locally free. We then pick a homomorphism

η : ι_{a}^{∗}Ob_{M/N} → K_{a}^{0} (4.12)

so that the composite K_{a}^{0} → ι_{a}^{∗}Ob_{M/N} → K_{a}^{0} is the identity. (Such lift exists possibly after shrinking Y_{a}^{0} ⊂ Y if necessary.) We let V_{1/2,a}^{0} = Vect(ι_{a}^{∗}Ob_{M/N}), let V_{a}^{0} = Vect(K_{a}^{0} ⊕ ι_{a}^{∗}Ob_{M}) and let \overline{η}:V_{1/2,a}^{0} → V_{a}^{0} be the immersion induced by η and the tautological Ob_{M/N} → Ob_{M}.

We now construct the intrinsic representative of a as a cycle in F_{a}^{0} × P^{1}, where F_{a}^{0} = V_{1/2,a}^{0} ×_{Y_{a}^{0}} V_{a}^{0} is a vector bundle stack over Y_{a}^{0}. Let pr : W_{α} → R_{α} be the projection. It is easy to see that for some dense open subset U of pr(T_{a}) ⊂ R_{a} we can lift the tautological U → Y_{a} to g : U → Y_{a}^{0} and lift the left vertical arrow (below) to surjective horizontal arrow (the top one) as shown in the commutative diagram

[L_{α} → E_{α}] ⊗_{O_{Rα}} O_{U} ⟶^{lift} g^{∗}[ι_{a}^{∗}Ob_{M/N} → K_{a}^{0} ⊕ ι_{a}^{∗}Ob_{M}]
↓  ↓
g^{∗}[Ob_{M/N} → Ob_{M/N}] ⟶ g^{∗}[ι_{a}^{∗}Ob_{M/N} → ι_{a}^{∗}Ob_{M/N}].

(4.13)

Note that the other two arrows are tautological ones. We let

(W_{α} × P^{1})|_{U×P^{1}} ⟶ F_{a}^{0} × P^{1} (4.14)

be the induced projection. Then an easy argument shows that the statement in Lemma 4.4 implies that there is a reduced and irreducible cycle Q_{a,η}^{0} ⊂ F_{a}^{0}×P^{1} so that the restriction of T_{a} to fibers over U ×P^{1} is a dense open subset of the flat pullback of Q_{a,η}^{0} via the arrow in (4.14). We call Q_{a,η}^{0} an intrinsic representative of a.

Once we have constructed the intrinsic representative of a ∈ Ξ(Q_{α}), we can define an equivalence relation on ∪Ξ(Q_{α}) as we did before. Let a ∈ Ξ(Q_{α}) and b ∈ Ξ(Q_{β}) be two elements. In case Y_{a} ≠ Y_{b}, then a ≁ b. In case Y_{a} = Y_{b}, we let Q_{a,η}^{0} and Q_{b,η}^{0} be the respective representatives of a and b, based on the same η in (4.12). Then we say a ∼ b if Q_{a,η}^{0}∩Q_{b,η}^{0} is dense in both Q_{a,η}^{0} and Q_{b,η}^{0}. This defines an equivalence relation. Further, whenever a ∼ b then m_{a} = m_{b}. Now let Ξ(Q) be the set of equivalence classes. For any a ∈ Ξ(Q), we pick a projective variety Y_{a} and a generically finite morphism φ_{a}:Y_{a} → Y_{a}. We then pick a pair of locally free subsheaves L_{a} ↪ E_{a} over Y_{a}, a surjective homomorphism of complexes

[L_{a} → E_{a}] ⟹ φ_{a}^{∗}[Ob_{M/N} → Ob_{M}]

and a lift (of the above arrow)

[L_{a} → E_{a}] ⟹ φ_{a}^{∗}[Ob_{M/N} → K_{a}^{0} ⊕ Ob_{M}]

over a dense open subset Y_{a}^{0} ⊂ Y_{a}. After that, we let W_{a} = Vect(L_{a}⊕E_{a}) and let j_{a} : W_{a}|_{Y_{a}^{0}} → F_{a}^{0} be the induced morphism. By shrinking Y_{a}^{0} if necessary, we can assume that W_{a}|_{Y_{a}^{0}} → F_{a}^{0} is flat. With all these chosen, we let Q_{a} be the closure in W_{a} × P^{1} of the flat pullback of Q_{a,η}^{0} and define ξ(a) = deg(φ_{a})^{−1}φ_{a∗}0_{W_{a}}^{!}[Q_{a}]. We define

[Q] = ∑_{a∈Ξ(Q)} m_{a}ξ(a) ∈ A_{∗}M × P^{1}.

It remains to prove the identity

∂_{∞}[Q] = ∑_{a∈Ξ(Q)} m_{a}∂_{∞}ξ(a) = [B_{2}] ∈ A_{∗}M (4.15)

and the similar identity with ∂_{∞} (resp. [B_{2}]) replaced by ∂_{0} (resp. [B_{1}]). We will prove the identity (4.15). The proof of the other identity is similar. To achieve this, we will define a function μ : Ξ(Q)×Ξ(B_{2}) → ℚ that satisfies

∑_{a∈Ξ(Q)} m_{a}μ(a, b) = m_{b}, ∀ b ∈ Ξ(B_{2}) (4.16)

and

∂_{∞}ξ(a) = ∑_{b∈Ξ(B_{2})} μ(a, b)ξ(b), ∀ a ∈ Ξ(Q). (4.17)

Once these two are established, then

∂_{∞}[Q] = ∑ m_{a} ∂_{∞}ξ(a) = ∑ m_{a} ∑ μ(a, b) ξ(b) = ∑ m_{b} ξ(b) = [B_{2}],

which is (4.15).

We first construct the function μ. We begin with any (smooth) chart R_{α}. We let L_{α} ⊕ E_{α} be the locally free sheaves on R_{α} chosen before. For simplicity we denote by W_{α} the sheaf L_{α} ⊕ E_{α} and continue to denote by W_{α} the vector bundle Vect(L_{α} ⊕ E_{α}). We let B_{2,α} ⊂ W_{α} and Q_{α} ⊂ W_{α} × P^{1} be the cycles constructed before. For each a ∈ Ξ(Q) we let Q_{α,a} be the closure in W_{α} × P^{1} of the pullback of Q_{a,η}^{0} via the arrow in (4.14). The cycle Q_{α,a} is also the union of irreducible components of Q_{α} in the equivalence class a. With this choice of Q_{α,a}, we have the decomposition

Q_{α} = ∑_{a∈Ξ(Q)} m_{a} Q_{α,a}. (4.18)

Similarly, we have a canonical decomposition

B_{2,α} = ∑_{b∈Ξ(B_{2})} m_{b} B_{2,α,b}.

As before, we let Ξ(B_{2,α}) be the index set of irreducible components of B_{2,α}. For c ∈ Ξ(B_{2,α}), we denote by T_{c} the corresponding component. Then because ∂_{∞}Q_{α} = B_{2,α}, there is a unique function μ_{α}(a, ·) : Ξ(B_{2,α}) → ℤ so that ∂_{∞}Q_{α,a} = ∑ μ_{α}(a, c)T_{c}. Again, by cycle (rational equivalence) consistency criteria and the invariance of the cycle under Aut(p), we conclude that whenever c_{1} ∼ c_{2} then μ_{α}(a, c_{1}) = μ_{α}(a, c_{2}). Thus μ_{α}(a, ·) descends to a function μ_{α}(a, ·) from Ξ(B_{2}) to ℤ. Finally, for b ∈ Ξ(B_{2}) we define B_{2,α,b} to be the union of T_{c} for those c ∈ Ξ(B_{2,α}) such that c ∼ b. Then we have the identity

∂_{∞}Q_{α,a} = ∑_{b∈Ξ(B_{2})} μ_{α}(a, b) B_{2,α,b}.

For the same reason whenever R_{α} ×_{M} R_{β} ×_{M} Y_{b} ≠ ∅, then μ_{α}(a, b) = μ_{β}(a, b). Hence, we can define μ(a, b) to be μ_{α}(a, b) for those α so that R_{α} ×_{M} Y_{b} ≠ ∅. Then the identity (4.16) follows from collecting term B_{2,α,b} in

∑_{b∈Ξ(B_{2})} m_{b} B_{2,α,b} = B_{2,α} = ∂_{∞}Q_{α} = ∑_{a∈Ξ(Q} m_{a} ∑_{b∈Ξ(B_{2})} μ(a, b) B_{2,α,b}

for some α so that R_{α} ×_{M} Y_{b} ≠ ∅.

We now investigate (4.17). Let a ∈ Ξ(Q) be any element, let φ_{a} : Y_{a} → Y_{a} be the generically finite morphism and let L_{a} ⊂ E_{a} be the pair of sheaves chosen before. We still denote by W_{a} the vector bundle Vect(L_{a} ⊕ E_{a}). We let Q_{a} ∈ W_{∗}W_{a} be the representative of a and let ∂_{∞}Q_{a} = ∑ m_{i}D_{i} be the decomposition into irreducible components.

Since Q_{a} → M factor through Y_{a} ⊂ M, all D_{i} lie over Y_{a}. We now let A ⊂ Y_{a} be a closed integral substack. We let Υ_{A} be (the index set of) those D_{i} so that the image D_{i} → M is exactly A. Similarly we let Ξ(B_{2})_{A} be those b ∈ Ξ(B_{2}) so that Y_{b} = A. Clearly if we can show that

deg(φ_{a})^{−1} ∑_{i∈Υ_{A}} m_{i} φ_{a∗}0_{W_{a}}^{!}[D_{i}] = ∑_{b∈Ξ(B_{2})_{A}} μ(a, b) ξ(b), (4.19)

since A ⊂ Y_{a} is arbitrary, (4.17) will follow immediately.

In the remainder of this section we will prove the identity (4.19) for any pair (a, A). We pick an étale φ_{U} : U → Y_{a} so that U ×_{Y_{a}} A → A is dominant. We then pick a pair of locally free sheaves L_{U} ⊂ E_{U} and surjective homomorphisms ϕ_{1} and ϕ_{2} of complexes as shown that make the diagram

[L_{U} → E_{U}]  ⟶^{ϕ_{1}} φ_{U}^{∗}[ι_{a}^{∗}Ob_{M/N} → K_{a}^{0} ⊕ ι_{a}^{∗}Ob_{M}]
↓^{ϕ_{2}}  ↓
φ_{U}^{∗}[Ob_{M/N} → Ob_{M}] ⟶ φ_{U}^{∗}[ι_{a}^{∗}Ob_{M/N} → ι_{a}^{∗}Ob_{M}]

commutative. Here the top-right corner is the complex in (4.13) and the right vertical arrow is the standard projection. As before, we let W_{U} be the vector bundle Vect(L_{U} ⊕ E_{U}). Then the homomorphism ϕ_{1} defines a flat morphism W_{U}|_{φ_{U}^{−1}(Y_{a}^{0})} → F_{a}^{0}. We let Q_{U} ⊂ W_{∗}W_{U} be the closure of the flat pullback of Q_{a,η}^{0} ∈ W_{∗}F_{a}^{0}.

We next consider the projections

q_{1} :U ×_{Y_{a}} Y_{a} → U and q̃_{1} :W_{U} ×_{Y_{a}} Y_{a} → W_{U}.

We let W̃ ⊂ W_{U} ×_{Y_{a}} Y_{a} be a dense open subset so that the tautological W̃ → Y_{a} factor through Y_{a}^{0} and W̃ → F_{a}^{0} is quasi-finite. We then let Q̃_{U} be the closure in (W_{U} ×_{Y_{a}} Y_{a}) × P^{1} of the flat pullback of Q_{a,η}^{0} via the obvious W̃ → F_{a}^{0}. Then since Q_{a,η}^{0} is dominant over Y_{a}^{0} and since q̃_{1} is proper, we have q̃_{1∗}(Q̃_{U}) = deg(φ_{a})Q_{U} and then

q̃_{1∗}(∂_{∞}Q̃_{U}) = ∂_{∞}q̃_{1∗}(Q̃_{U}) = deg(φ_{a})∂_{∞}Q_{U}. (4.20)

For the convenience of the readers, we list the related rational equivalence relations constructed:

Q_{a,η}^{0} ∈ W_{∗}F_{a}^{0}, Q_{U} ∈ W_{∗}W_{U}, Q̃_{U} ∈ W_{∗}(W_{U} ×_{Y_{a}} Y_{a}) and Q′ ∈ W_{∗}W.

(Q′ will be constructed in (4.23).) We now let Υ (resp. Υ̃) be the (index) set of those irreducible components of ∂_{∞}Q_{U} (resp. ∂_{∞}Q̃_{U}) that lie over and dominate A. For i ∈ Υ or Υ̃ we denote by T_{i} the corresponding component and by m_{i} the multiplicity of Q_{U} or Q̃_{U} along T_{i}. We need to divide Υ̃ into two parts: One is Υ̃_{1} which consists of those T_{i} so that q̃_{1∗}(T_{i}) ≠ 0. We let Υ̃_{0} = Υ̃ − Υ̃_{1}. Because of (4.20), there is a map λ: Υ̃_{1} → Υ so that under q̃_{1∗} the component T_{i} is mapped onto T_{λ(c)}, say via ψ_{i,λ(i)} :T_{i} → T_{λ(i)}. Then (4.20) implies

∑_{i∈λ^{−1}(j)} deg(ψ_{i,j}) = deg(φ_{a}), j ∈ Υ. (4.21)

We now compare the collection Υ̃ with the collection Υ_{A}. We let p_{1} and p_{2} be the projections of U ×_{Y_{a}} Y_{a} to U and Y_{a}, respectively. We pick a pair of locally free sheaves L ↪ E on U ×_{Y_{a}} Y_{a} so that there are two surjective homomorphisms ϕ_{1} and ϕ_{2} of complexes as shown in the commutative diagram

\xymatrix{[L → E] \ar[r]^{ϕ_{1}} \ar[d]_{ϕ_{2}} & p_{2}^{∗}[L_{a} → E_{a}] \ar[d] \\ p_{1}^{∗}[L_{U} → E_{U}] \ar[r] & [ι_{a}^{∗}Ob_{M/N} → K_{a}^{0} ⊕ ι_{a}^{∗}Ob_{M}] \ar[r] & K_{a}^{0} ⊕ Ob_{M}|_{Y_{a}^{0}}].} (4.22)

Here the two remaining arrows in the diagram are the ones chosen before. We let W be the vector bundle Vect(L ⊕ E) over U ×_{Y_{a}} Y_{a} and let ζ_{1} : W → W_{U} ×_{Y_{a}} Y_{a} and ζ_{2} : W → W_{a} be the morphisms to vector bundles over U ×_{Y_{a}} Y_{a} and Y_{a} induced by ϕ_{1} and ϕ_{2}, respectively. The map ζ_{1} is obviously smooth whose fibers are vector spaces. Since U ×_{Y_{a}} Y_{a} → Y_{a} is étale, ζ_{2} is also smooth with affine fibers. Further, because of the commutative diagram (4.22) and the rational equivalence consistency criteria, the flat pullback of Q_{U} via ζ_{1} and the flat pullback of Q_{a} via ζ_{2} are identical over a dense open subset U ⊂ U ×_{Y_{a}} Y_{a} that is flat over both U and Y_{a}. Hence since both Q_{U} and Q_{a} are closed and since all their irreducible components dominate Y_{a},

Q′ ≜ ζ_{1}^{∗}(Q_{U}) = ζ_{2}^{∗}(Q_{a}). (4.23)

We now let Υ′ be the set of those irreducible components of ∂_{∞}Q′ that lie over and dominate A. Since the fibers of ζ_{1} are vector spaces, Υ′ is naturally identical to Υ̃, say via λ′: Υ′ → Υ̃. Further, for each i ∈ Υ′ the component T_{i} is a vector bundle over T_{λ′(i)}. On the other hand, since p_{2} :U ×_{Y_{a}} Y_{a} → Y_{a} is smooth, each irreducible component of ∂_{∞}Q′ is a component of the flat pullback of a T_{i} for some i ∈ Υ_{A}. Thus there is a map λ_{a}:Υ′ → Υ_{A} so that for each i ∈ Υ_{A} the union ∑_{j∈λ_{a}^{−1}(i)} T_{j} is exactly the pullback of T_{i}. For i ∈ Υ′ and j ∈ λ_{a}(i), we let φ_{i,j}:T_{i} → T_{j} be the tautological map.

We are now ready to prove the identity (4.19). For each i ∈ Υ′, we let b(T_{i}) be the image of T_{i} → U ×_{Y_{a}} Y_{a}. We pick a projective variety Z_{i} so that a dense open subset Z_{i}^{0} ⊂ Z_{i} is a finite branched cover of b(T_{i}) and the induced Z_{i}^{0} → A extends to φ_{i}:Z_{i} → A. We then pick a pair of locally free sheaves L_{i} ⊂ E_{i} over Z_{i} and a surjective homomorphism of complexes ϕ′_{1} that lifts to a surjective ϕ′_{2} as shown in the commutative diagram

[L_{i} → E_{i}]  ⟶^{ϕ′_{2}}  [L|_{Z_{i}^{0}} → E|_{Z_{i}^{0}}]
↓^{ϕ′_{1}}  ↓
φ_{i}^{∗}[Ob_{M/N} → Ob_{M}] ⟶ φ_{i}^{∗} [Ob_{M/N}|_{Z_{i}^{0}} → Ob_{M}|_{Z_{i}^{0}}] .

Then following the definition of ξ we let W_{i} = Vect(L_{i} ⊕ E_{i}) and let C_{i} ⊂ W_{i} × P^{1} be the closure of the flat pullback of T_{i} ⊂ W |_{Z_{i}^{0}} × P^{1} via the induced W_{i}|_{Z_{i}^{0}} → W |_{Z_{i}^{0}}. We then define (the non-normalized) \overline{ξ}(i) = φ_{i∗}0_{W_{i}}^{!}[C_{i}] ∈ A_{∗}M × P^{1}.

For i ∈ Υ, Υ̃ or Υ_{A} we define the variety Z_{i}^{0} to be the image of T_{i} in U, in U ×_{Y_{a}} Y_{a} or Y_{a} respectively, and then define φ_{i}:Z_{i} → A and the class \overline{ξ}(i) along the same line. We let ψ_{U,A} be the induced morphism U ×_{Y_{a}} A → A. We have the following lemma concerning the relations λ:Υ̃_{1} → Υ, λ′:Υ′ → Υ̃ and λ_{a}:Υ′ → Υ_{A} defined before.

Lemma 4.5. The following relations hold:

1. For each i ∈ Υ′ we have \overline{ξ}(i) = \overline{ξ}(λ′(i)).

2. For each i ∈ Υ̃ we have deg(φ_{i,λ_{a}(i)}) \overline{ξ}(λ_{a}(i)) = \overline{ξ}(i) and for any j ∈ Υ_{A} we have ∑_{i∈λ_{a}^{−1}(j)} deg(φ_{i,j}) = deg(ψ_{U,A}).

3. For any i ∈ Υ̃_{1} we have deg(ψ_{i,λ(i)}) \overline{ξ}(λ(i)) = \overline{ξ}(i).

4. For i ∈ Υ̃_{0} we have \overline{ξ}(i) = 0.

Proof. The proof of (1) is parallel to the proof of Lemma 2.6, using the fact that the fibers of ζ_{1} are vector spaces. We will omit the proof here. The proof of the identity in (2) concerning \overline{ξ}(·) is similar and will be omitted too. The identity concerning the degrees in (2) follows from the fact that U ×_{Y_{a}} Y_{a} → Y_{a} is étale. We now prove (3). Let i ∈ Υ̃ and j = λ(i) ∈ Υ. We let Z_{i}^{0} ⊂ Z_{i} and Z_{j}^{0} ⊂ Z_{j} be the pair of varieties constructed in defining the classes \overline{ξ}(i) and \overline{ξ}(j). By definition, there is a canonical dominant morphism Z_{i}^{0} → Z_{j}^{0}. Hence without loss of generality we can assume that it extends to ρ:Z_{i} → Z_{j}. With this choice of Z_{i} and Z_{j}, we can choose the pairs of sheaves L_{i} ⊂ E_{i} and L_{j} ⊂ E_{j} be so that the former is the pullback of the later via ρ. Then because of the relation (4.20), the cycle representatives C_{i} ∈ Z_{∗}Vect(L_{i} ⊕ E_{i}) and C_{j} ∈ Z_{∗}Vect(L_{j} ⊕ E_{j}) satisfies ρ̃_{∗}C_{i} = deg(ψ_{i,j})C_{j}. Here we used the fact that C_{i} is supported on a single variety. This relation implies (3) immediately. The proof of (4) is parallel and will be omitted. This proves the lemma. q.e.d.

It follows from the lemma that

deg(ψ_{U,A}) ∑_{i∈Υ_{A}} m_{i} 0_{W_{a}}^{!}[D_{i}] = deg(ψ_{U,A}) ∑_{i∈Υ_{A}} m_{i} \overline{ξ}(i) = ∑_{j∈Υ′} m_{j} \overline{ξ}(j).

Here we used (2) to derive the last identity. Because \overline{ξ}(i) = 0 for i ∈ Υ̃_{0}, by (1) of the lemma the right hand side above is

∑_{i∈Υ̃_{1}} m_{i} \overline{ξ}(i) = ∑_{i∈Υ̃_{1}} deg(ψ_{i,λ(i)}) m_{i} \overline{ξ}(λ(i)) = deg(φ_{a}) ∑_{i∈Υ} m_{i} \overline{ξ}(i)

= deg(φ_{a}) deg(ψ_{U,A}) ∑_{b∈Ξ(B_{2})_{A}} μ(a, b)ξ(b).

Here the first equality follows from (3) in the lemma, the second equality follows from (4.21) and the last identity follows from the definition of μ(a, b), using the fact that U is étale over Y_{a}. This proves the identity (4.19) and hence completes the proof of Lemma 4.3. q.e.d.

4.2 Applications

In this subsection, we will study three cases of relative obstruction theories and derive some identities.

We first study the case where N is a DM-stack and M → N is a substack defined by the vanishing of a C-divisor (L, s). Namely if S_{α} → N is a chart (in an atlas Λ) and (L_{α}, s_{α}) is the associated Cdivisor on S_{α}, then M×_{N}S_{α} is defined by the vanishing of s_{α}. We denote M×_{N} S_{α} by R_{α}. We claim that there is a canonical relative obstruction theory of R_{α}/S_{α} taking values in the sheaf O_{R_{α}}(L_{α}) (= O_{R_{α}}(L_{α}|_{R_{α}})). Since R_{α} is a subscheme of S_{α}, the relative first order deformations 𝔇𝔢𝔣_{R_{α}/S_{α}}^{1} = 0. Now let ξ = (B, I, φ) be any object in 𝔗𝔯𝔦_{R_{α}/S_{α}}. Since B is a Γ(O_{S_{α}})-algebra, its associated morphism ϕ: Spec B → S_{α} is an extension of Spec B/I → S_{α}. We consider the section s_{α} ∘ ϕ of ϕ^{∗}L_{α}. Since s_{α}∘ϕ|_{Spec B/I} ≡ 0 ∈ Γ(φ^{∗}L_{α}), s_{α}∘ϕ is an element in Γ(R_{α}, L_{α})⊗I. We define this element to be the relative obstruction class 𝔬𝔟_{R_{α}/S_{α}}(ξ). Of course, 𝔬𝔟_{R_{α}/S_{α}}(ξ) = 0 if and only if ϕ:Spec B → S_{α} factor through R_{α} ⊂ S_{α}. Since R_{α} → S_{α} is an immersion, φ extends to Spec B → R_{α} as S_{α}-morphism if and only if ϕ already factor through R_{α} ⊂ S_{α}. This proves that 𝔬𝔟_{R_{α}/S_{α}} is an obstruction assignment. Since (L, s) is a Cdivisor on N, this defines a relative obstruction theory of M/N.

Now assume both M and N have perfect obstruction theories provided by the data {E_{α}^{•}, 𝔬𝔟_{R_{α}}}_{Λ} and {F_{α}^{•}, 𝔬𝔟_{R_{α}/S_{α}}}_{Λ}. We assume further that the obstruction theories of M and N are compatible to the relative obstruction theory of M/N in the sense of Definition 4.1 with L_{α}^{•} = [0 → O_{R_{α}}(L_{α})].

Lemma 4.6.

Let the notation be as before. Then [M]^{virt} = c_{1}(L, s)[N]^{virt}.

Proof. We will follow the notation developed before and after the Lemma 4.2. Let p ∈ N be any point, S_{α} be an étale chart of N with a lift \overline{p} ∈ S_{α} of p. We let T_{2} = h^{1}(F_{α}^{•} ⊗ 𝕜\overline{p}) and let O_{2} = h^{2}(F_{α}^{•} ⊗ 𝕜\overline{p}). We let Ŵ_{p} be the formal completion of S_{α} along \overline{p}. Here we use subscript p instead \overline{p} since Ŵ_{p} depends on p up to Aut(p). We know Ŵ_{p} is Spec 𝕜[[T_{2}^{∨}]]/(g), where g is a Kuranishi map of the obstruction theory of S_{α} at \overline{p}. We denote by C_{p} the normal cone to Ŵ_{p} in W_{p} ≜ Spec 𝕜[[T_{2}^{∨}]]. The cone C_{p} is naturally embedded in O_{2} × Ŵ_{p}. Again, the pair C_{p} ⊂ O_{2} × Ŵ_{p} only depends on the point p ∈ N, up to the symmetry Aut(p). By the construction of the virtual moduli cycles in [26] and in Section 2.2, this collection of cone cycles {C_{p}}_{p∈N} can be algebraicized and hence gives rise to a cycle, the virtual moduli cycle [N]^{virt}.

Now assume p ∈ M. Then \overline{p} ∈ R_{α} where R_{α} = S_{α} ×_{N} M. As in (4.3), we let T_{1} = h^{1}(E_{α} × 𝕜\overline{p}) and O_{1} = h^{2}(E_{α} ⊗ 𝕜\overline{p}), etc. There are two cases to consider. One is when δ : T_{2} → O_{1/2} in (4.3) is 0. In this case T = T_{1} = T_{2}, O = O_{1} and O_{2} = O_{1}/O_{1/2}. Further, we can choose the relative Kuranishi map h ∈ 𝕜[[T^{∨}]]/(g) ⊗ O_{1/2} to be ŝ_{α}, the pullback of s_{α} via Spec 𝕜[[T^{∨}]]/(g) ≅ Ŵ_{p} → S_{α}. Here we have used the canonical isomorphism O_{1/2} = L_{α}|\overline{p}. Now let X_{p} and X̂_{p} be the schemes Spec 𝕜[[T^{∨}]] and Spec[[T^{∨}]]/(f), where f is the Kuranishi map in (4.5). Then X_{p} = W_{p} and the cone D(p)_{2} is the normal cone to C_{p} ×Ŵ_{p} {ŝ_{α} = 0} in C_{p}. The cone D(p)_{2} is naturally embedded in V̂_{1/2}×X̂_{p} V̂_{2}. For the same reason, when δ is nonzero, thus surjective since dim O_{1/2} = 1, the cone D(p)_{2} is also the normal cone to C_{p}×Ŵ_{p} {ŝ_{α} = 0} in C_{p}.

We are now ready to prove the lemma. First, from the discussion before Lemma 4.3 we know the collection {D(p)_{2}}_{p∈M} can be algebraicized. By definition, the cycle constructed based on {D(p)_{2}} following the basic construction in Section 2.2 is the relative virtual cycle [M, N]^{virt}. However, since ŝ_{α} are pullback of the section s_{α}, the cone D(p)_{2} is the normal cone to C_{p} ×_{S_{α}} {s_{α} = 0} in C_{p}. Further, since the collection {(L_{α}, s_{α})} is the restriction of (L, s) to charts S_{α}, a repetition of the proof of Lemma 4.3 shows that [M, N]^{virt} = c_{1}(L, s)[N]^{virt}. Then combined with the identity [M]^{virt} = [M, N]^{virt} in Lemma 4.3, we have [M]^{virt} = c_{1}(L, s)[N]^{virt}. This completes the proof of the lemma. q.e.d.

We now investigate the second case. We let N be a DM-stack with a morphism N → X to a scheme X. For simplicity we assume X is smooth. Let ξ : X_{0} ⟶ X be a smooth subvariety and let M be defined by the Cartesian product

M = N ×_{X} X_{0}.

Then M is a substack of N. Let L over M be the pull back of the normal bundle to X_{0} in X. Similar to the case just studied, there is a canonical relative obstruction theory of M/N taking values in the cohomology of the complex L^{•} = [0 → O_{M}(L)]. We now assume M and N both have perfect obstruction theories and are compatible to the relative obstruction theory of M/N. By the intersection theory of DM-stacks [38], the Gysin map η^{!}[N]^{virt} ∈ A_{∗}M.

Lemma 4.7. Let the notation be as above. Then [M, N]^{virt} = η^{!}[N]^{virt}.

Proof. The proof is similar to that of the previous lemma with slight modification. We shall omit the proof here. q.e.d.

We now investigate the third case. We let Q be a smooth Artin stack with k C-divisor (L_{i}, s_{i}) and k positive integers n_{i}, i = 1, …, k.

We define N ⊂ Q be the substack defined by the vanishing of the sections s_{1}^{n_{1}}, …, s_{k}^{n_{k}}. Then if we let \overline{ρ}_{α}:\overline{S}_{α} → Q be a smooth chart and let S_{α} = \overline{S}_{α} ×_{Q} N, then S_{α} has a natural perfect obstruction theory taking values in the complex

F_{α}^{•} = [O_{S_{α}}(T \overline{S}_{α}|_{S_{α}}) ⟶ ⊕_{i}O_{S_{α}}(ρ_{α}^{∗}L_{i}^{⊗n_{i}})], (4.24)

where ρ_{α}: S_{α} → N is the tautological morphism. We now let M be a DM-stack over N and let R_{α} → M be a chart so that R_{α} → M → N factor through R_{α} → S_{α}. We assume M has a perfect obstruction theory given by {E_{α}^{•}, 𝔬𝔟_{R_{α}}}_{Λ}, where Λ is the atlas {R_{α}}_{Λ} of M. We also assume M/N has a perfect relative obstruction theory given by {L_{α}^{•}, 𝔬𝔟_{R_{α}/S_{α}}}. Finally, we assume all these obstruction theories are compatible. Hence by Lemma 4.3 we have [M]^{virt} = [M, N]^{virt}.

What we are interested is to compare this cycle with the virtual moduli cycle of the substack M_{0} ⊂ M defined by M_{0} = M ×_{N} N_{0}, where N_{0} ⊂ Q is defined by the vanishing of the sections s_{1}, …, s_{k}. Note that M is homeomorphic to M_{0}. Again we assume M_{0} has perfect obstruction theory so that it is compatible to the perfect obstruction theory of N_{0} and the perfect relative obstruction theory of M_{0}/N_{0}. For simplicity, we only consider the case where N_{0} is smooth and Codim(N_{0}, Q) = k.

Lemma 4.8. Suppose the relative obstruction theory of M_{0}/N_{0} is induced from that of M/N. Then [M, N]^{virt} = (∏_{i=1}^{k} n_{i}) [M_{0}, N_{0}]^{virt}.

Here by the relative obstruction theory of M/N inducing a relative obstruction theory of M_{0}/N_{0} we mean that the relative obstruction sheaf Ob_{M_{0}/N_{0}} = Ob_{M/N} ⊗_{O_{M}} O_{M_{0}} and the relative obstruction class assignment 𝔬𝔟_{M_{0}/N_{0}}(ξ) = 𝔬𝔟_{M/N}(ξ) for any triple ξ ∈ Ob(𝔗𝔯𝔦_{M_{0}/N_{0}}).

Proof. Without loss of generality, we can assume all n_{i} ≥ 2 since otherwise we can replace Q by Q ∩ \{s_{i} = 0|n_{i} = 1\}. Let \overline{S}_{α} → Q, S_{α} = \overline{S}_{α} ×_{Q} N and let R_{α} → M be as before. We let (L_{i,α}, s_{i,α}) be the restriction of (L_{i}, s_{i}) to \overline{S}_{α}. Then S_{α} = \overline{S}_{α} ∩ \{s_{i,α}^{n_{1}} = ⋯ = s_{i,α}^{n_{k}} = 0}. We let L_{α}^{•} and E_{α}^{•} be the complexes of sheaves over R_{α} that are part of the (relative) obstruction theories of M/N and M as stated in Definition 4.1. As before, we let F_{α}^{•} be the complex (4.24). Again, we assume that the exact sequences (4.1) hold. We now let \overline{p} ∈ R_{α} be any closed point and let \overline{q} ∈ S_{α} be the image of \overline{p}. We let T_{i} and O_{i} (i = 1/2, 1, 2 or ∅) be the vector spaces defined before (4.3) associated to \overline{p} and the pair R_{α}/S_{α}. Since all n_{i} ≥ 2, the vector space O_{2} is ⊕_{i=1}^{k}L_{i,α}|\overline{p}. Also we can choose the Kuranishi map g_{p} ∈ 𝕜[[T_{2}^{∨}]] ⊗ O_{2} to be the germ of s_{α}^{[n]} = (s_{1,α}^{n_{1}}, …, s_{k,α}^{n_{k}}). We let W = Spec 𝕜[[T_{2}]] and let Ŵ, X and X̂ be the schemes defined after Lemma 4.2. Then since Codim(S_{α}, \overline{S}_{α}) = k and since \overline{S}_{α} is smooth, the cone C_{Ŵ /W} (which is defined after (4.7)) is the vector bundle O_{2} × Ŵ over Ŵ. In this case the corresponding germ D(p)_{2}, which is the normal cone to C_{Ŵ /W} ×_{W} X̂ in C_{Ŵ /W} ×_{W} X, is C_{Ŵ /W} ×Ŵ C_{X̂×_{W} Ŵ /X×_{W} Ŵ}.

Now we consider the parallel situation for the pair M_{0} → N_{0}. We let S_{0,α} = \overline{S}_{α} ×_{Q} N_{0} and let R_{0,α} = R_{α} ×_{Q} N_{0}. We will denote by \overline{p} ∈ R_{0,α} and \overline{q} ∈ S_{0,α} the same points \overline{p} ∈ R_{α} and \overline{q} ∈ S_{α}, via the inclusion R_{0,α} ⊂ R_{α} and S_{0,α} ⊂ S_{α}. We then let T_{0,i} and O_{0,i} be the vector spaces defined before (4.3) associated to the pair R_{0,α}/S_{0,α} over the point \overline{p}. By assumption, T_{0,2} ⊂ T_{2}, T_{0,1/2} ≡ T_{1/2}, T_{0} ⊂ T is a codimension k linear subspace, O_{0,2} = 0 and O_{0,1/2} = O_{1/2}. Thus we have the following diagram

0 ⟶ O_{1/2} ⟶ O ⟶ O_{2} ⟶ 0
∥  ↓  ↓
0 ⟶ O_{0,1/2} ⟶ O_{0} ⟶ 0.

Further, we can choose the residue of h ∈ 𝕜[[T^{∨}]]⊗O_{1/2} in 𝕜[[T_{0}^{∨}]]⊗O_{1/2} ≡ 𝕜[[T_{0,1/2}^{∨}]] ⊗ O_{0,1/2}, denoted by h_{0}, be the relative Kuranishi map of R_{0,α}/S_{0,α} at \overline{p}. We let W_{0}, Ŵ_{0}, X_{0} and X̂_{0} be the similarly defined formal schemes associated to \overline{q} ∈ S_{0,α} and \overline{p} ∈ R_{0,α}. Note that with this choice, the Kuranishi map g_{0} = 0 and Ŵ_{0} = W_{0}. Hence the associated germ D(p)_{0,2} is the normal cone C_{X̂_{0}/X_{0}}, which is a cycle in O_{0,1/2} ×X̂_{0}.

Since Ŵ_{0} is Ŵ with the reduced scheme structure and since the relative Kuranishi map h_{0} is the restriction of the Kuranishi map h to Ŵ_{0}, Z_{0} = Z ×_{W} W_{0} and Ẑ_{0} = Ẑ ×Ŵ W_{0}. From this we see that the cycle D(p)_{2} is a multiple of the pull-back of D(p)_{0,2} under the projection

(O_{1/2} × O_{2}) × Ẑ ⟶^{proj} O_{0,1/2} × Ẑ

with the multiplier given by the multiplicity of Ŵ along Ŵ_{0}, which is ∏_{i=1}^{k} n_{i}.

Based on this, we see that Ξ(M/N) is canonically isomorphic to Ξ(M_{0}/N_{0}). Further, for any a ∈ Ξ(M/N) with the corresponding \overline{a} ∈ Ξ(M_{0}/N_{0}), a representative (A_{a}, F_{a}, φ_{a}) of a is also a representative of \overline{a}. Of course their multiplicities obey m_{a} = m\overline{a} ∏ n_{i}. Therefore

[M/N]^{virt} = ∑_{a∈Ξ(M/N)} m_{a} ξ(a) = ∑_{a∈Ξ(M_{0}/N_{0})} m_{a} ∏_{i=1}^{k} n_{i} · ξ(a)

= (∏_{i=1}^{k} n_{i}) [M_{0}/N_{0}]^{virt}.

This proves the lemma. q.e.d.

4.3 Proof of Lemmas 3.10 and 3.11

The proof of Lemma 3.10 is similar to that of Lemma 3.11 while technically less involved. Hence we will prove Lemma 3.11 and omit the other. The strategy to prove Lemma 3.11 is to apply Lemma 4.6 to the case where M = 𝔐(𝔚_{0}, η) and N = 𝔐(𝔚, Γ). To this end, we need to work our the relative obstruction theory of 𝔐(𝔚_{0}, η)/𝔐(𝔚, Γ).

Following the argument in Section 2.1, we only need to look at the relative obstruction theory of 𝔐(W_{0}[n], η)^{st}/𝔐(W [n], Γ)^{st}. We first cover 𝔐(W [n], Γ)^{st} by affine étale charts S_{α} → 𝔐(W [n], Γ)^{st}. We let (L_{η,α}, s_{η,α}) be the restriction of (L_{η}, s_{η}) to S_{α}. By definition, R_{α} ≜ S_{α} ×_{𝔐(W [n],Γ)^{st}} 𝔐(W_{0}[n], η)^{st} is the subscheme of S_{α} defined by the vanishing of s_{α}. Hence R_{α}/S_{α} admits an obvious relative obstruction theory induced by the pair (L_{η,α}, s_{η,α}), as defined in the first case in Subsection 4.2. Namely, for any ξ = (B, I, φ) ∈ 𝔗𝔯𝔦_{R_{α}/S_{α}} with φ:Spec B/I → R_{α}, the relative obstruction class is

𝔬𝔟_{R_{α}/S_{α}}(ξ) = d(s_{α} ∘ φ) ∈ Γ(R_{α}, L_{η,α}) ⊗_{Γ(R_{α})} I.

Hence the relative obstruction theory of R_{α}/S_{α} takes values in the cohomology of the complex [0 → O_{R_{α}}(L_{η,α})]. Because L_{η,α} are the restriction of a global line bundle L_{η} on 𝔐(W [n], Γ)^{st} and the sections s_{α} over S_{α} are the restrictions of the global section s_{η}, the collection of the relative obstruction assignments {𝔬𝔟_{R_{α}/S_{α}}} are compatible over R_{αβ} and thus defines a global relative obstruction theory of 𝔐(W_{0}[n], η)^{st}/𝔐(W [n], Γ)^{st} taking values in the cohomology of

[0 → O_{𝔐(W_{0}[n],η)^{st}}(L_{η})].

We now let A_{α} = Γ(O_{S_{α}}) and A_{η,α} = Γ(O_{R_{α}}). Without lose of generality, we can assume all S_{α} are η-admissible (cf. Definition 3.3).

We then form the complex of A_{α}-modules E_{α}^{•} as in (1.19) and the complex of A_{η,α}-modules E_{η,α}^{•} as in (3.10), with the subscript α added to emphasize their dependence on α. Recall that they are the respective complexes that are part of the perfect obstruction theories of S_{α} and R_{α}. Further, we have the exact sequence of complexes (3.11). Thus to apply Lemma 4.6 we only need to show that the relative obstruction theory of 𝔐(W_{0}[n], η)^{st}/𝔐(W [n], Γ)^{st} is compatible to the obstruction theories of 𝔐(W_{0}[n], η)^{st} and of 𝔐(W [n], Γ)^{st}, in the sense of Definition 4.1. Because of the exact sequence (3.11), we only need to show that for any triple ξ = (B, I, φ) in 𝔗𝔯𝔦_{R_{α}/S_{α}} we have

ζ(𝔬𝔟_{R_{α}/S_{α}}(ξ)) = 𝔬𝔟_{R_{α}}(ξ).

Here ζ is the homomorphism h^{1}(C^{•−1}(O_{R_{α}}(L_{η,α}))) → h^{2}(E_{η,α}^{•}(O_{R_{α}})). But this follows directly from the construction of the respective (relative) obstruction theories. As argued before, the relative obstruction theory of 𝔐(W_{0}[n], η)^{st}/𝔐(W [n], Γ)^{st} descends to a relative obstruction theory of 𝔐(𝔚_{0}, η)/𝔐(𝔚, Γ) taking values in the cohomology of the complex [0 → O_{𝔐(𝔚_{0},η)}(L_{η})], and this relative obstruction theory is compatible to the obstruction theories of 𝔐(𝔚_{0}, η) and 𝔐(𝔚, Γ). Thus by applying the result proved in Subsection 4.2, we conclude [𝔐(𝔚_{0}, η)]^{virt} = c_{1}(L_{η}, s_{η})[𝔐(𝔚, Γ)]^{virt}. This proves Lemma 3.11.

4.4 Proof of Lemma 3.12

We now prove Lemma 3.12. Here is our strategy. Let M_{g,n} be the (Artin) stack of k-pointed genus g nodal curves and let

μ : 𝔐(𝔚, Γ) ⟶ M_{g,n}

be the forgetful morphism. Let η = (Γ_{1}, Γ_{2}, I) ∈ Ω be as in Lemma 3.12 that has r ordered roots of weights μ_{1}, …, μ_{r}. We will show that in the formal neighborhood of 𝔐(𝔚_{0}, η) in 𝔐(𝔚, Γ) (possibly after an étale base change) there are divisors (L_{i}, s_{i}) for 1 ≤ i ≤ r so that 𝔐(𝔚_{0}, η) is defined by the vanishing of s_{1}^{μ_{1}}, …, s_{r}^{μ_{r}} while 𝔐(𝔜_{1}^{rel} ⊔ 𝔜_{2}^{rel}, η) is defined by the vanishing of s_{1}, …, s_{r}. This way we can reduce the proof of Lemma 3.12 to the situation studied in Lemma 4.8.

We now provide the detail of the proof. We first construct the desired base change of the formal neighborhood of 𝔐(𝔚_{0}, η) in 𝔐(𝔚, Γ). We let M_{Γ_{i}^{o}} be the moduli stack of pointed nodal curves (not necessary connected) of topological type Γ_{i}^{o} (see the definition before (2.9)). Since we do not impose stability condition on such curves, M_{Γ_{i}^{o}} is an Artin stack. Then for any pair (C_{1}, C_{2}) ∈ M_{Γ_{1}^{o}} ×M_{Γ_{2}^{o}} we can form a new curve C_{1} ⊔ C_{2} by gluing the pairs of the i-th distinguished marked point of C_{1} and of C_{2} for all i = 1, …, r. This construction extends to families, thus defines a (local embedding) morphism M_{Γ_{1}^{o}} × M_{Γ_{2}^{o}} → M_{g,k}.

Definition 4.9. Let A be an Artin stack. We say B is a formal extension of A if A is a closed substack of B and the inclusion A → B is a homeomorphism.

Lemma 4.10. We can find a formal extension Q of M_{Γ_{1}^{o}} × M_{Γ_{2}^{o}} so that the morphism M_{Γ_{1}^{o}} × M_{Γ_{2}^{o}} → M_{g,k} extends to an étale morphism Q → M_{g,k}.

Proof. For schemes, this is the topological equivalence of étale morphisms [30]. Note that once such extensions exist, then they are canonical. The proof of the general case can be proved by applying this topological equivalence theorem to charts of the stacks. We will leave the details to the readers. q.e.d.

We next consider the gluing morphism

Φ_{η} : 𝔐(𝔜_{1}^{rel}, Γ_{1}) ×_{D^{r}} 𝔐(𝔜_{1}^{rel}, Γ_{1}) ⟶ 𝔐(𝔜_{1}^{rel} ⊔ 𝔜_{2}^{rel}, η) ⊂ 𝔐(𝔚, Γ).

Lemma 4.11. There is a formal extension 𝔐(𝔚, Γ)ˆ of 𝔐(𝔜_{1}^{rel}, Γ_{1}) ×_{D^{r}}𝔐(𝔜_{1}^{rel}, Γ_{1}) (as DM-stack) so that the morphism Φ_{η} extends to an étale morphism

Φ̂_{η} : 𝔐(𝔚, Γ)ˆ ⟶ 𝔐(𝔚, Γ).

Proof. The proof is similar to Lemma 4.10, and will be omitted. q.e.d.

Because 𝔐(𝔚, Γ)ˆ is homeomorphic to 𝔐(𝔜_{1}^{rel}, Γ_{1})×_{D^{r}} 𝔐(𝔜_{1}^{rel}, Γ_{1}), the forgetful morphism

(4.25) 𝔐(𝔜_{1}^{rel}, Γ_{1}) ×_{D^{r}} 𝔐(𝔜_{2}^{rel}, Γ_{2}) ⟶ M_{Γ_{1}^{o}} × M_{Γ_{2}^{o}}

extends to 𝔐(𝔚, Γ)ˆ → Q.

Our next task is to define the PD-divisors (L_{i}, s_{i}) on Q for i = 1, …, r as mentioned. We let ξ and ζ be the universal families over M_{g,k} and M_{Γ_{1}^{o}} × M_{Γ_{2}^{o}}. We let ξ_{node} be the natural substack of all nodal points of the fibers of ξ. Then ξ_{node} ⊂ ξ is a smooth divisor. On the other hand, each fiber ζ_{p} (over p ∈ M_{Γ_{1}^{o}} × M_{Γ_{2}^{o}}) contains r ordered distinguished nodes. They define r ordered distinguished sections

n_{1}, …, n_{r} : M_{Γ_{1}^{o}} × M_{Γ_{2}^{o}} ⟶ ζ. (4.26)

The nodal locus ζ_{node} ⊂ ζ is the union of a smooth divisor with the images of these r sections. We now consider the universal family \overline{ζ} over Q. Since Q → M_{g,k} is étale, the nodal locus \overline{ζ}_{node} is a smooth divisor in \overline{ζ}. On the other hand, since \overline{ζ}|_{M_{Γ^{o}_{1}}×M_{Γ^{o}_{2}}} ≅ ζ, ζ_{node} ⊂ \overline{ζ}_{node} and is a homeomorphism. Hence each Im(n_{i}) is an open subset in \overline{ζ}_{node}. We let \overline{B}_{i} ⊂ \overline{ζ}_{node} be the open substack that contains and is homeomorphic to Im(n_{i}). We let B_{i} ⊂ Q be the image stack of \overline{B}_{i} under the projection \overline{ζ} → Q. Clearly, B_{i} are smooth divisors of Q. We define (L_{i}, s_{i}) be the C-divisor on Q so that B_{i} = s_{i}^{−1}(0).

For later application, we now give trivializations of (L_{i}, s_{i}) on charts of Q. let T_{α} → Q be any chart. Then B_{i,α} = T_{α} ×_{Q} B_{i} is a smooth divisor in T_{α}. Without loss of generality, we can assume that B_{i,α} is defined by the vanishing of a u_{i,α} ∈ Γ(O_{T_{α}}). We then choose L_{i,α} be the line bundle over T_{α} so that O_{T_{α}}(L_{i,α}) = u_{i,α}^{−1}O_{T_{α}} and let s_{i,α} ∈ Γ(L_{i,α}) be the constant 1. Put it differently, e_{i,α} = u_{i,α}^{−1}1 is a global holomorphic basis of L_{i,α} while the section s_{i,α} = 1 = u_{i,α}e_{i,α} vanishes on B_{i,α}. In case ũ_{i,α} is another defining equation of B_{i,α}, we define (L̃_{i,α}, s̃_{i,α}) similarly via a basis ẽ_{i,α} = u_{i,α}^{−1}1 and s̃_{i,α} = ũ_{i,α}ẽ_{i,α}. The transition function is via ẽ_{i,α} = (u_{i,α}/ũ_{i,α})e_{i,α}.

Now let μ_{i} be the weight of the i-th root of η. We define

N_{0} = {s_{1} = ⋯ = s_{r} = 0} and N = {s_{1}^{μ_{1}} = ⋯ = s_{r}^{μ_{r}} = 0} ⊂ Q, (4.27)

both are substacks of Q. Clearly, N_{0} = M_{Γ_{1}^{o}} × M_{Γ_{2}^{o}}. Hence

𝔐(𝔜_{1}^{rel} ⊔ 𝔜_{2}^{rel}, η) = 𝔐(𝔚, Γ)ˆ ×_{Q} N_{0}.

We define

𝔐(𝔚_{0}, η)^{et} = 𝔐(𝔚, Γ)ˆ ×_{Q} N.

Lemma 4.12. Let π_{Q} : 𝔐(𝔚, Γ)ˆ → Q and π_{M} : 𝔐(𝔚, Γ)ˆ → 𝔐(𝔚, Γ) be the tautological projections. Then we have isomorphisms of C-divisors

π_{Q}^{∗}(L_{i}, s_{i})^{⊗μ_{i}} ≅ π_{M}^{∗}(L_{η}, s_{η}).

Further, 𝔐(𝔚_{0}, η)^{et} → 𝔐(𝔚, Γ) factor through an étale 𝔐(𝔚_{0}, η)^{et} → 𝔐(𝔚_{0}, η).

Proof. We cover 𝔐(𝔚, Γ)ˆ by an atlas of étale charts S_{α} →𝔐(𝔚, Γ)ˆ indexed by Λ. To each α ∈ Λ, we let f_{α}: X_{α} → W [n_{α}] be the pullback of the universal family over 𝔐(𝔚, Γ). We then cover Q by charts T_{α} indexed by the same set Λ. Without loss of generality, we can assume that the composite of S_{α} → 𝔐(𝔚, Γ)ˆ → Q factors through h_{α}: S_{α} → T_{α} ⊂ Q and the family X_{α} is the pullback of the universal family \overline{ζ}_{α} on T_{α}. As before, we let B_{i,α} = T_{α} ×_{Q} B_{i} ⊂ T_{α} and let u_{i,α} be a defining equation of B_{i,α}; We let (L_{i,α}, s_{i,α}) be the restriction of (L_{i}, s_{i}) to the chart T_{α}. As mentioned, we can choose e_{i,α} = u_{i,α}^{−1}1 to be a holomorphic basis of L_{i,α}, and hence s_{i,α} = u_{i,α}e_{i,α}. Now we let n_{i,α}: B_{i,α} → \overline{ζ}_{α}|_{B_{i,α}} be the lift of the section n_{i} in (4.26) to B_{i,α}. Since Q is homeomorphic to M_{Γ_{1}^{o}} × M_{Γ_{2}^{o}}, such a lift exists and is unique. We then let N_{i,α} = n_{i,α}(B_{i,α}) and let N̂_{i,α} be the formal completion of \overline{ζ}_{α} along N_{i,α}. Without loss of generality, we can assume

N̂_{i,α} ≅ Spec 𝕜[[z_{1}, z_{2}]] ×_{Spec 𝕜[t]} T_{α}.

Here Spec 𝕜[[z_{1}, z_{2}]] → Spec 𝕜[t] is defined by t ↦ z_{1}z_{2} and T_{α} → Spec 𝕜[t] is defined by t ↦ u_{i,α}.

We now back to the family f_{α}:X_{α} → W [n_{α}]. Recall X_{α} = \overline{ζ}_{α} ×_{Q} S_{α}. We let X̂_{i,α} be the formal completion of X_{α} along N_{i,α} ×_{Q} S_{α}, which is isomorphic to X_{α} ×_{\overline{ζ}_{α}} N̂_{i,α}. By shrinking S_{α} if necessary, we can assume that there is a parameterization of a neighborhood of nodes W_{α} ⊂ W [n_{α}] given by

ψ_{α} : W_{α} ⟶ Spec 𝕜[w_{1}, w_{2}] ⊗_{𝕜[t_{lα}]} Γ(A^{n_{α}+1})

for some l_{α} ∈ [n_{α} + 1], as in (1.1), so that the induced morphism f̂_{α}: X̂_{i,α} → W [n_{α}] factors through

f̃_{α}:X̂_{i,α} → W_{α} via f̃_{α}^{∗}(w_{j}) = β_{j,α} · z_{j}^{μ_{j}}, j = 1, 2. (4.28)

Now let (L_{η,α}, s_{η,α}) be the restriction to S_{α} of (L_{η}, s_{η}). By definition, a trivialization of (L_{η,α}, s_{η,α}) is given by O(L_{η,α}) = t_{l_{α}}^{−1}O_{T_{α}} with the basis ϵ_{η,α} = t_{l_{α}}^{−1}1 and s_{η,α} ≡ t_{l_{α}}ϵ_{η,α}. As mentioned, a trivialization of (L_{i,α}, s_{i,α}) is given by O(L_{i,α}) = u_{i,α}^{−1}O_{S_{α}} with e_{i,α} = u_{i,α}^{−1}1 and s_{i,α} = u_{i,α}e_{i,α}. We then define an isomorphism L_{η,α} ≅ L_{i,α}^{⊗μ_{i}} via

ϵ_{η,α} = (β_{α,1}β_{α,2})e_{i,α}^{⊗μ_{i}}.

Note that since β_{α,1}β_{α,2} ∈ Γ(O_{S_{α}}^{×}), the above identity defines an isomorphism L_{η,α} ≅ L_{i,α}^{⊗μ_{i}}. Further, because of the relations (4.28) and w_{1}w_{2} = t_{l_{α}},

s_{η,α} = s_{i,α}^{⊗μ_{i}} (4.29) under this isomorphism. Hence it defines an isomorphism of the corresponding C-divisors. It is routine to check that the so defined isomorphisms extend to an isomorphism of C-divisors

π_{Q}^{∗}(L_{η}, s_{η}) ≅ π_{M}^{∗}(L_{i}, s_{i})^{⊗μ_{i}}.

This proves the isomorphisms of the C-divisors.

The last statement follows directly from the construction of 𝔐(𝔚_{0}, η)^{et}. q.e.d.

We now consider the pair ρ: 𝔐(𝔚_{0}, η)^{et} → N. Since 𝔐(𝔚_{0}, η)^{et} is étale over 𝔐(𝔚_{0}, η), we can take the obstruction theory of 𝔐(𝔚_{0}, η)^{et} to be the pullback of that of 𝔐(𝔚_{0}, η). We now workout the relative obstruction theory of 𝔐(𝔚_{0}, η)^{et}/N and to show that it satisfies the set up in Lemma 4.8. We will follow closely the notation developed in the beginning of Section 4.1. For convenience, we denote 𝔐(𝔚_{0}, η)^{et} by M.

We begin with a smooth chart \overline{S} of Q and the associated chart S = \overline{S} ×_{Q} N. We let (L_{i}, s_{i}) be the restriction of (L_{i}, s_{i}) to \overline{S}. Since Q and hence \overline{S} are smooth, the pairs (L_{i}, s_{i}) for i = 1, …, r define a natural obstruction theory of S taking values in the cohomology of the complex

F^{•} = [O_{S}(T \overline{S}) ⟶^{ds^{[μ]}} ⊕_{i=1}^{r} O_{S}(L_{i}^{⊗μ_{i}})] , (4.30)

as defined in the beginning of the Section 4.1. Here ds^{[μ]} is the abbreviation of (ds_{1}^{μ_{1}}, …, ds_{r}^{μ_{r}}). Now let X be the universal family over S and let D ⊂ X be the divisor of the marked sections of X. Then by the deformation theory of nodal curves, there is a canonical homomorphism of sheaves (the Kodaira map)

O_{S}(T \overline{S}) ⟶ Ext _{X /S}^{1}(Ω_{X /S}(D), O_{X}).

Next we pick an affine étale universal open R → S ×_{N} M with ρ:R → S the projection. Without loss of generality we can assume that the universal family of R is of the form f : ρ^{∗}X → W [n] for some integer n. We now construct the standard obstruction theory of R. Since R is a smooth chart (not necessary étale) of M, its obstruction theory is slightly different from that defined in Section 3.1, which is for étale charts of M.

We begin with the construction of the complex E_{η}^{•} that will be part of the obstruction theory of R. We let B = Γ(O_{R}) with A = Γ(O_{S}) as before. First of all, we choose the complex F^{•} in (1.15) to be

F^{0} = 0 and F^{1} = Γ(R, ρ^{∗}T \overline{S}) = Γ(O_{S}(T \overline{S})) ⊗_{A} B.

We pick a collection of charts (U_{α}/V_{α}, f_{α}) of f indexed by Λ, as mentioned in the paragraph before the equation (1.17). By shrinking R and/or make an étale base change of \overline{S}, we can assume that there is a collection of charts {U_{α}/V_{α}} of X /S indexed by the same Λ so that V_{α} = R×_{S} V_{α} and U_{α} = R×_{S} U_{α}. We form the modules Hom_{U_{·}}(f^{∗}Ω_{W [n]}, B)^{†}, after picking the necessary data as shown in the paragraphs before the equation (1.17). Since we have chosen F^{0} = 0, the homomorphism ζ_{α}(·) in (1.18) is zero. The homomorphism ζ_{αβ}(·) is exactly the one defined in (1.17). We then form the complex D^{•}, the homomorphism δ and the complex E^{•}, following the definitions after Lemma 1.14, line by line except that we replace F^{0} by 0. To form the complex E_{η}^{•}, we shall follow the discussion after (3.8). We let C_{η}^{•} be the complex defined in (3.7) and let δ:E^{•} ⇒ C_{η}^{•−1} be the homomorphism defined exactly as in (3.8). We then let E_{η}^{•} be the complex defined in (3.10). It follows from the discussion before that the obstruction theory of M induces a natural obstruction theory of R taking values in the cohomology of the complex E_{η}^{•}.

We now construct a complex F^{•} of A-modules that is quasi-isomor-phic to F^{•} (in (4.30)) and a homomorphism of complexes E_{η}^{•} ⇒ F^{•}. Let i be any integer between 1 and r. We let Λ_{i} ⊂ Λ be the subset of the indices α so that U_{α} → X contains the i-th distinguished nodes of some fibers of X /S. Then the collection {U_{α}}_{α∈Λ_{i}} forms a covering of a neighborhood of the i-th distinguished nodes of the fibers of X /S and the collection {V_{α}}_{Λ_{i}} forms an étale covering of S. We let C_{i}^{k} = C^{k}(Λ_{i}, L_{i}^{⊗μ_{i}}) be the Cěch complex of k-cochains of the line bundle L_{i}^{⊗μ_{i}} associated to the covering {V_{α}}_{Λ_{i}}. The complex C_{i}^{k} comes with the standard coboundary operator C_{i}^{k} → C_{i}^{k+1}. We define F^{1} = Γ(O_{S}(T \overline{S})) and define F^{k} = ⊕_{i=1}^{r}C_{i}^{k−2} for k ≥ 2.

We next define the differential ∂^{k}: F^{k} → F^{k+1}. Let I be the ideal sheaf of the zero section of the total space T \overline{S} and let \overline{S}^{(2)} be the subscheme of T \overline{S} defined by the ideal sheaf I^{2}. Then there is a tautological morphism κ: \overline{S}^{(2)} → \overline{S} characterized by the following property: Let v ∈ T_{p}\overline{S} with Spec 𝕜[t]/(t^{2}) → T_{p}\overline{S} its associated morphism that lifts to the morphism [v] : Spec 𝕜[t]/(t^{2}) → \overline{S}^{(2)}, then the composite κ ∘ [v] : Spec 𝕜[t]/(t^{2}) → \overline{S} is the tangent vector v ∈ T_{p}\overline{S}. Now let ξ ∈ Γ(O_{S}(T \overline{S})) be any element. We let S̃ = Spec Γ(O_{S}) ∗ A be the trivial extension of S by A and let θ_{ξ} : S̃ → \overline{S} be the composite S̃ →

\overline{S}^{(2)} ⟶^{κ} \overline{S}, where S̃ → \overline{S}^{(2)} is defined by ξ. Since θ_{ξ}^{∗}(s_{i}^{μ_{i}}) ∈ Γ(θ_{ξ}^{∗}L_{i}^{⊗μ_{i}})

and θ_{ξ}^{∗}(s_{i}^{μ_{i}})|_{S} ≡ 0,

d[θ_{ξ}^{∗}(s_{i}^{μ_{i}}) − 0] ∈ Γ(S, O_{S}(L_{i}^{⊗μ_{i}}) ⊗ I_{S⊂S̃}) = Γ(S, O_{S}(L_{i}^{⊗μ_{i}})).

Here I_{S⊂S̃} is the ideal of S ⊂ S̃, which is isomorphic to A. Now let ρ_{α}^{∗}θ_{ξ}^{∗}(s_{i}^{μ_{i}}) ∈ Γ(V_{α}, O_{S}(L_{i}^{⊗μ_{i}})) be the pullback under ρ_{α} : V_{α} → S. We then define

∂_{i}^{1}(ξ)_{α} = d[ρ_{α}^{∗}θ_{ξ}^{∗}(s_{i}^{μ_{i}}) − 0] ∈ Γ(V_{α}, O_{S}(L_{i}^{⊗μ_{i}})), α ∈ Λ_{i},

and define ∂^{1} = ⊕_{i}∂_{i}^{1}. For k ≥ 1 we let ∂^{k}:F^{k} → F^{k+1} to be the direct sum of the coboundary operators of C_{i}^{•}. Clearly, the so defined operator ∂^{•} satisfies ∂^{k} ∘ ∂^{k+1} = 0, and hence defines a complex F^{•} = (F^{k}, ∂^{k}). Further, it follows from our construction that the complex F^{•} is quasiisomorphic to the complex F^{•} in (4.30).

We now define the promised homomorphism φ^{•} : E_{η}^{•} ⇒ F^{•} ⊗_{A} B. Recall that E_{η}^{1} = Γ(O_{S}(T \overline{S})) ⊗_{A} B ⊕ D^{0} and F^{1} = Γ(O_{S}(T \overline{S})). The homomorphism φ^{1} : E_{η}^{1} → F^{1} ⊗_{A} B is the one induced by the identity of Γ(O_{S}(T \overline{S})) ⊗_{A} B. For k ≥ 2, we notice that E_{η}^{k} = E^{k} ⊕ C_{η}^{k−2} and F^{k} = ⊕_{i=1}^{r}C_{i}^{k−2}. The homomorphism φ^{k} will be induced by φ_{i}^{k} : C_{η}^{k} → C_{i}^{k} ⊗_{A} B, which we define now. Let ξ ∈ C_{η}^{k} be any element and let (α_{0} … α_{k}) be a (k + 1)-tuple in Λ_{i}. Then ξ_{α_{0}...α_{k}} is an element in Γ(V_{α_{0}...α_{k}}, L_{η}). Here L_{η} is the restriction of L_{η} to R. Using the canonical isomorphism L_{η} ≅ ρ^{∗}L_{i}^{⊗μ_{i}}, where ρ:R → S is the tautological projection,

ξ_{α_{0}...α_{k}} ∈ Γ(V_{α_{0}...α_{k}}, ρ^{∗}L_{i}^{⊗μ_{i}}) = Γ(V_{α_{0}...α_{k}}, L_{i}^{⊗μ_{i}}) ⊗_{A} B.

We denote this element by ξ̃_{α_{0}...α_{k}}. We define

φ_{i}^{k}(ξ)_{α_{0}...α_{k}} = ξ̃_{α_{0}...α_{k}} ∈ Γ(V_{α_{0}...α_{k}}, L_{i}^{⊗μ_{i}}) ⊗_{A} B.

This defines a homomorphism C_{η}^{k} → ⊕_{i=1}^{r}C_{i}^{k} ⊗_{A} B.

We claim that the so defined homomorphisms form a homomorphism of complexes E_{η}^{•} ⇒ F^{•} ⊗_{A} B. For this, we need to check the commutativity of the following diagram

\begin{matrix}E_{η}^{k} & ⟶^{d^{k}} & E_{η}^{k+1} \\ ^{φ^{k}}↓ & & ^{φ^{k+1}}↓ \\ F^{k} ⊗_{A} B & ⟶^{∂^{k}} & F^{k+1} ⊗_{A} B. \end{matrix} (4.31)

We will prove the commutativity of (4.31) for the case k = 1. The other cases are similar and will be omitted. First, we show that (4.31) commutes on D^{0} ⊂ E_{η}^{1}. Let ξ ∈ D^{0} ⊂ E_{η}^{1} be any element given by a collection {ξ_{α}} of ξ_{α} ∈ Hom_{U_{α}}(f^{∗}Ω_{W [n]}, O_{S})^{†}, as defined in (1.11). Because of the relation (1.12), the composite D^{0} ⊂ E_{η}^{1} → E_{η}^{2} ⟶^{φ^{2}} F^{2} ⊗_{A} B is zero. On the other hand, by definition we have φ^{1}(ξ) = 0. Hence φ^{2}(∂^{1}(ξ)) = d^{1}(φ^{1}(ξ)) for all ξ ∈ D^{0}.

We next check that (4.31) commutes on Γ(O_{S}(T \overline{S})) ⊂ E_{η}^{1}. We first recall the definition of φ^{2} ∘ d^{1}. Let ξ ∈ Γ(O_{S}(T \overline{S})) be any element. It determines a morphism κ̃_{ξ}:R̃ = Spec Γ(O_{R})∗B → \overline{S} that is the pullback of κ_{ξ}:S̃ → \overline{S} via the tautological extension ρ̃:R̃ → S̃ of ρ:R → S. We let α ∈ Λ_{i} be any index with U_{α}/V_{α} the associated chart of the universal family X /S and with Ũ_{α}/Ṽ_{α} its minimal extension to the family X̃/S̃. We let U_{α}/V_{α} and Ũ_{α}/Ṽ_{α} be the corresponding pullback charts of ρ^{∗}X /R and ρ^{∗}X̃/R̃. We then pick a local parameterization of the nodes of U_{α}/V_{α} and its extension to Ũ_{α}/Ṽ_{α}. We let (z_{α,i}, s_{α}) and (z̃_{α,i}, s̃_{α}) be the relevant functions associated to these parameterizations^{20} . We let the parameterizations of the nodes of U_{α}/V_{α} and Ũ_{α}/Ṽ_{α} be the pull back of those from U_{α}/V_{α} and Ũ_{α}/Ṽ_{α}. We let f_{α}:U_{α} → W [n] be the restriction of f to U_{α} and let f̃_{α} : Ũ_{α} → W [n] be a predeformable extension of f_{α}. We let ı_{α} : U_{α} → U_{α}, ĩ: Ũ_{α} → Ũ_{α}, ȷ_{α} : V_{α} → V_{α}, j̃_{α} : Ṽ_{α} → Ṽ_{α}, ρ_{α}: U_{α} → V_{α} and ρ̃_{α}: U_{α} → V_{α} be the tautological projections. Then after picking a local parameter of f_{α}(U_{α}) ⊂ W [n], say (w_{1}, w_{2}) with w_{1}w_{2} = t_{l_{α}}, we have

f_{α}^{∗}(w_{α,i}) = ı_{α}^{∗}(z_{α,i}^{μ_{i}}) · h_{α,i} and f̃_{α}^{∗}(w_{α,i}) = ı_{α}^{∗}(z_{α,i}^{μ_{i}}) · h̃_{α,i}

for some h_{α,i} ∈ Γ(O_{U_{α}}^{×}) and their extensions h̃_{α,i} ∈ Γ(O× ; Ũ_{α}) that satisfy

h_{α,1}h_{α,2} ∈ Γ(O_{V_{α}}) and h̃_{α,1}h̃_{α,2} ∈ Γ(OṼ_{α}). Since z̃_{α,1}z̃_{α,2} = s̃_{α}, we have

f̃_{α}^{∗}(t_{l_{α}}) = (h̃_{α,1}h̃_{α,2})ı_{α}^{∗}(s_{α}^{μ_{i}}).

20By abuse of notation we will view s_{α} as functions on V_{α} and on U_{α} via the pull back O_{V_{α}} → O_{U_{α}}. The same convention applies to s̃_{α} as well.

Then

φ^{2}(d^{1}(ξ))_{α} = d[f̃_{α}^{∗}(t_{l_{α}}) − 0]
= (h_{α,1}h_{α,2})d[ȷ_{α}^{∗}(s_{α}^{μ_{i}}) − 0] ∈ Γ(V_{α}, ρ^{∗}L_{η,α} ⊗ I_{R⊂R̃}).

Here the last relation follows because f̃_{α} is predeformable.

As to d^{1} ∘ φ^{1}, by definition

d^{1}(φ^{1}(ξ))_{α} = d[s_{α}^{μ_{i}} − 0] ∈ Γ(V_{α}, ρ^{∗}L_{i}^{⊗μ_{i}} ⊗ I_{S⊂S̃}).

Therefore we have φ^{2}(∂^{1}(ξ))_{α} = d^{1}(φ^{1}(ξ))_{α} because of the isomorphism L_{η} ≅ L_{i}^{⊗μ_{i}}, (4.31), the relation t_{l_{α}} = (h_{α,1}h_{α,2})s_{α}^{μ_{i}} and I_{R⊂R̃} ≅ B and T_{S⊂S̃} ≅ A. This proves the commutativity of (4.31) for k = 1. The case k ≥ 2 is similar and will be omitted.

We define G^{k} = ker{E_{η}^{k} → F^{k} ⊗_{A} B}. Since E_{η}^{k} → F^{k} ⊗_{A} B is surjective, G^{k} is a flat O_{S}-module. The differentials of E_{η}^{•} induces differentials of G^{•} and the resulting complex fits into the following exact sequence

0 ⟹ G^{•} ⟹ E_{η}^{•} ⟹ F^{•} ⊗_{A} B ⟹ 0.

It is routine to check that for each ξ = (B′, I, φ_{0}) ∈ 𝔗𝔯𝔦_{S/R} there is a canonical obstruction class 𝔬𝔟_{R/S}(ξ) ∈ h^{2}(G^{•} ⊗ I) to extending φ_{0} : Spec B′/I → R to an S-morphism Spec B′ → R, and further such assignment satisfies the requirement in Definition 4.1. Finally, we remark that though the complexes G^{•}, E_{η}^{•} and F^{•} depend on the choice of the covering {U_{α}/V_{α}} of f, they as elements in the derived category are unique. In particular the modules (sheaves) Ob_{R/S} ≜ h^{2}(G^{•}), Ob_{R} ≜ h^{2}(E_{η}^{•}) and Ob_{S} ≜ h^{2}(F^{•}) are independent of the choice of the coverings.

We now cover N be an atlas {S_{α}}_{Ξ} and for each α ∈ Ξ we pick an open étale R_{α} → M ×_{N} S_{α} so that {R_{α}}_{Ξ} forms an atlas of M. For each α we pick a sufficiently fine covering of its universal family and then form the associated complexes G_{α}^{•}, E_{η,α}^{•} and E_{α}^{•}. Here we added the subscript α to indicate the dependence on the chart R_{α}/S_{α}. To be consistent with the notation in Definition 4.1, we let L_{α}^{•} = G_{α}^{•}, E_{α}^{•} = E_{η,α}^{•} and F_{α}^{•} = F_{α}^{•}, viewed as complexes of sheaves of O_{R_{α}} or O_{S_{α}}-modules accordingly.

Lemma 4.13. There are standard relative obstruction theories of R_{α}/S_{α} for α ∈ Ξ taking values in the complexes L_{α}^{•} such that the (relative) obstruction theories {E_{α}^{•}, 𝔬𝔟_{R_{α}}}, {F_{α}^{•}, 𝔬𝔟_{S_{α}}} and {L_{α}^{•}, 𝔬𝔟_{R_{α}/S_{α}}} are compatible in the sense of Definition 4.1. Further, the so defined obstruction theories of R_{α}/S_{α} define a relative obstruction theory of M/N that is compatible to the obstruction theories of M and N.

Proof. The proof is routine and will be omitted. q.e.d.

Let N_{0} ⊂ N be as in (4.27) and let M_{0} = M ×_{N} N_{0}. By definition M_{0} is an étale cover of 𝔐(𝔜_{1}^{rel} ⊔ 𝔜_{2}^{rel}, η). Note that N_{0} is smooth. As to M_{0}, we endow it with the induced obstruction theory of 𝔐(𝔜_{1}^{rel} ⊔ 𝔜_{2}^{rel}, η), which is perfect since that of 𝔐(𝔜_{1}^{rel}⊔𝔜_{2}^{rel}, η) is. We let [M_{0}]^{virt} be the virtual moduli cycle of M_{0}. Then Lemma 3.12 is equivalent to

(4.32) 𝔪(η)[M_{0}]^{virt} = [M]^{virt} ∈ A_{∗}M.

By Lemma 4.8, to prove this identity it suffices to show that:

1) There is a relative obstruction theory of M_{0}/N_{0} that is compatible to the obstruction theory of M_{0} and N_{0}.

2) The relative obstruction theory M_{0}/N_{0} is compatible to the relative obstruction theory of M/N.

The proof of 1) is parallel to the construction of the relative obstruction theory of M/N. The proof of 2) is immediate once the relative obstruction theory was constructed. Since the proof is routine, we will leave it to the readers. We state it as a lemma.

Lemma 4.14. The standard relative obstruction theory of M_{0}/N_{0} is compatible to the obstruction theories of M_{0} and N_{0}. Further, the relative obstruction theory of M_{0}/N_{0} is induced from the obstruction theory of M/N.

In the end, we apply Lemma 4.8 to the pairs M_{0}/N_{0} ⊂ M/N to conclude (4.32). This completes the proof of Lemma 3.12.

4.5 Proof of Lemma 3.14

It remains to prove Lemma 3.14. Let

(4.33) Φ_{η} : 𝔐(𝔜_{1}^{rel}, Γ_{1}) ×_{D^{r}} 𝔐(𝔜_{2}^{rel}, Γ_{2}) ⟶ 𝔐(𝔜_{1}^{rel} ⊔ 𝔜_{2}^{rel}, η)

be the étale morphism in (3.2). Using the Cartesian product (3.14), we can give 𝔐(𝔜_{1}^{rel}, Γ_{1}) ×_{D^{r}} 𝔐(𝔜_{2}^{rel}, Γ_{2}) a canonical obstruction theory. We call such obstruction theory the obstruction theory induced by the Cartesian product. On the other hand, since Φ_{η} is étale, the obstruction theory of 𝔐(𝔜_{1}^{rel} ⊔ 𝔜_{2}^{rel}, η) induces an obstruction theory of 𝔐(𝔜_{1}^{rel}, Γ_{1}) ×_{D^{r}} 𝔐(𝔜_{2}^{rel}, Γ_{2}) as well.

Lemma 4.15. The two obstruction theories of 𝔐(𝔜_{1}^{rel}, Γ_{1}) ×_{D^{r}} 𝔐(𝔜_{2}^{rel}, Γ_{2}), one defined by the Cartesian product and the other induced by that of 𝔐(𝔜_{1}^{rel} ⊔ 𝔜_{2}^{rel}, η), are identical.

Proof. The proof is similar to that in [26], and will be omitted. q.e.d.

We now consider the virtual moduli cycle of 𝔐(𝔜_{1}^{rel}, Γ_{1})×_{D^{r}} 𝔐(𝔜_{2}^{rel}, Γ_{2}). Since the obstruction theory of 𝔐(𝔜_{1}^{rel}, Γ_{1})×_{D^{r}}𝔐(𝔜_{2}^{rel}, Γ_{2}) is induced by the Cartesian product,

[𝔐(𝔜_{1}^{rel}, Γ_{1}) ×_{D^{r}} 𝔐(𝔜_{2}^{rel}, Γ_{2})]^{virt}
= Δ^{!}([𝔐(𝔜_{1}^{rel}, Γ_{1})]^{virt} × [𝔐(𝔜_{2}^{rel}, Γ_{2})]^{virt}).

On the other hand, because of Lemma 4.15 and because Φ_{η} is étale of pure degree | Eq(η)|,

\frac{1}{| Eq(η)|}Φ_{η∗}Δ^{!}([𝔐(𝔜_{1}^{rel}, Γ_{1})]^{virt} × [𝔐(𝔜_{2}^{rel}, Γ_{2})]^{virt})
= [𝔐(𝔜_{1}^{rel} ⊔ 𝔜_{2}^{rel}, η)]^{virt}.

This is exactly Lemma 3.14.

This completes the proof of the degeneration formula of the Gromov-Witten invariants stated in the beginning of this paper.

5. Appendix

5.1 The tangent and the obstruction spaces

In this appendix, we will express the first order deformation and the obstruction spaces of 𝔐(𝔚, Γ) and 𝔐(ℨ^{rel}, Γ) in terms of some known cohomology groups. As a corollary, we will show that the obstruction theories we constructed in this paper are all perfect.

Let S → 𝔐(W [n], Γ)^{st} be an affine étale chart. As before, we denote by f:X → W [n] be the universal family with D ⊂ X the divisor of the ordinary marked sections. As before, we let π:X → S be the projection and let ρ : S → A^{n+1} be the morphism under f. By shrinking and making an étale base change, we can assume the following holds for S: For each l the projection induced by π

f^{−1}(D_{l})_{red} ⟶ (S ×_{A^{n+1}} H^{l})_{red}

is a union of r_{l} disjoint sections σ_{l,i}: (S ×_{A^{n+1}} H^{l})_{red} → f^{−1}(D_{l})_{red} for i = 1, …, r_{l}. (Note that r_{l} could be zero.) We pick an atlas (U_{α}/V_{α}, f_{α}) of f indexed by Λ so that for each l and i ∈ [1, r_{l}] there is exactly one and only one α so that U_{α} ∩ Im(σ_{l,i}) ≠ ∅, and hence Im(σ_{l,i}) is covered by U_{α}. As before, we let Λ_{l} be the collection of those α so that U_{α}/V_{α} covers Im(σ_{l,i}) for some i. When α is of the second kind, we let (z_{α,1}, z_{α,2}, s_{α}) be the parameterization of the distinguished nodes of U_{α}/V_{α} (cf. before Simplification 1.7). We require s_{α}^{m_{α}} = g^{∗}(t_{l}) in case α ∈ Λ_{l}. With such assumptions and choices made, the standard log structure of S is given by the prelog structure N_{S} = ⊕N_{l} → O_{S} given in (1.7).

We now let E^{•} and D^{•} be the complexes associated to the perfect obstruction theory of S constructed in Section 1. Let A = Γ(O_{S}). For simplicity, we give an ordering of Λ_{l} and thus the r_{l} charts in Λ_{l} are U_{l,1}/V_{l,1}, …, U_{l,r_{l}}/V_{l,r_{l}}. We let R_{l}^{•} be the complex

(s_{l,1}^{m_{l,1}}, …, s_{l,r_{l}}^{m_{l,r_{l}}}) : O_{S}^{⊕r_{l}} ⟶ (O_{S}^{⊕r_{l}}/O_{S}) ⊗_{O_{A^{n+1}}} O_{A^{n+1}}(H^{l}),

where O_{S}^{⊕r_{l}}/O_{S} is the quotient of O_{S}^{⊕r_{l}} by the diagonal O_{S} ↪ O_{S}^{⊕r_{l}}. In case r_{l} = 0, we agree R_{l}^{•} = [A → 0]. Here the complex R_{l}^{•} is indexed at [0, 1].

Recall that H^{1}(E^{•}) is the space of first order deformations of f while H^{2}(E^{•}) is the obstruction space to deforming f.

Proposition 5.1. For any A-module I, we have the following two exact sequences

0 ⟶ Ext_{X}^{0} (Ω_{X /S}(D), I) ⟶ H^{0}(D^{•} ⊗_{A} I) ⟶ H^{1}(E^{•} ⊗_{A} I)

⟶ Ext_{X}^{1} (Ω_{X /S}(D), I) ⟶ H^{1}(D^{•} ⊗_{A} I) ⟶ H^{2}(E^{•} ⊗_{A} I) ⟶ 0

and

0 ⟶ H^{0}(Hom(f^{∗}Ω_{W [n]^{†}/A^{n+1†}}, I)) ⟶ H^{0}(D^{•} ⊗_{A} I)

⟶^{b_{0}} ⊕_{l=1}^{n+1} H_{et}^{0}(R_{l}^{•} ⊗_{O_{S}} I) ⟶^{δ} H^{1}(Hom(f^{∗}Ω_{W [n]^{†}/A^{n+1†}}, I))

⟶ H^{1}(D^{•} ⊗_{A} I) ⟶^{b_{1}} ⊕_{l=1}^{n+1} H_{et}^{1}(R_{l}^{•} ⊗_{O_{S}} I) ⟶ 0.

Further, H^{i}(E^{•}) = 0 for i > 2 and H^{i}(D^{•}) = 0 for i ≥ 2. Here I = I ⊗_{A} O_{X}.

Here Ω_{W [n]^{†}/A^{n+1†}} is the sheaf of log differentials of the pair of log schemes (cf. [17, 18]). Also, the cohomology H_{et}^{i}(R_{l}^{•}) is the étale cohomology of the complex of locally relatively constant sections, namely, sheaves of O_{S}-modules over X_{et}.

Proof. The first exact sequence follows directly from the construction of the complexes D^{•} and E^{•}. We now prove the second exact sequence. We first construct the arrow b_{0}. Let ξ ∈ H^{0}(D^{•} ⊗ I) be any element. Then ξ is represented by {ξ_{α}} where ξ_{α} ∈ Hom_{U_{α}}(f^{∗}Ω_{W [n]}, I)^{†}. In case α ∈ Λ_{l} then

ξ_{α} = (φ_{α}, η_{α,1}, η_{α,2}) ∈ Hom_{U_{α}}(f^{∗}Ω_{W [n]}, I_{α}) ⊕ I_{α}^{⊕2}, I_{α} = I ⊗_{O_{X}} O_{U_{α}}, (5.1)

following the convention in (1.10). Specifically, the η_{·} means dw_{i}/w_{i} ↦ η_{α,i}, under the appropriate parameterization (w_{1}, w_{2}) of f_{α}(U_{α}). Recall that Λ_{l} is an ordered set, thus for α = i ∈ Λ_{l} we denote η_{(l,i)} = η_{α,1}+η_{α,2}. Then we define

b_{0}(ξ)_{l} = (η_{(l,1)}, …, η_{(l,r_{l})}) ∈ A^{⊕r_{l}}

and then the arrow b_{0} is

b_{0}(ξ) = (b_{0}(ξ)_{1}, …, b_{0}(ξ)_{l}).

In case r_{l} = 0, then we pick an α ∈ Λ and define η_{l} = φ_{α}(dt_{l}), where φ_{α} is part of the ξ_{α} as in (5.1). It is direct to check that if ξ is a cohomology class then b_{0}(ξ)_{l} ∈ H_{et}^{0}(R_{l}^{•} ⊗ I). This defines the corresponding arrow in the second exact sequence.

We next construct the arrow H_{et}^{1}(D^{•}) → H_{et}^{1}(R^{•}). Let Σ_{l} = f^{−1}(D_{l}) ⊂ X and let I_{Σ_{l}⊂X} be the relative locally constant ideal sheaf defined by

Γ(U_{α}, I_{Σ_{l}⊂X}) = \{ \begin{matrix}Γ(V_{α}, O_{V_{α}}) & in case U_{α} ∩ f^{−1}(D_{l}) = ∅ \\ s_{α}^{m_{α}}Γ(V_{α}, O_{V_{α}}) & otherwise. \end{matrix}

Similarly, we let π^{−1}O_{S} be the pullback sheaf, namely Γ(U_{α}, π^{−1}O_{S}) = Γ(V_{α}, O_{V_{α}}). Then we have the exact sequence in étale site

0 ⟶ I_{Σ_{l}⊂X} ⟶ π^{−1}O_{S} ⟶ ⊕_{α∈Λ_{l}}(O_{S}/s_{α}^{m_{α}}O_{S}) ⟶ 0

and its induced exact sequence in cohomologies (of O_{S}-modules)

0 ⟶ g^{∗}(t_{l})O_{S} → O_{S} ⟶ ⊕_{α∈Λ_{l}}(O_{S}/s_{α}^{m_{α}}O_{S}) ⟶ H^{1}(R_{l}^{•}) ⟶ 0.

Now we back to the complex D^{•}. It follows from the construction of D^{•} that there is a homomorphism of complexes D^{•} → C^{•}(Λ, I_{Σ_{i}⊂X}). The arrow b_{1} = ⊕_{l=1}^{n+1}b_{1,l}. The arrow δ in the second sequence is the ordinary connecting homomorphism. We leave it to readers to check that with these arrows the second sequence is exact.

The vanishing result stated in Lemma 1.19 follows from these exact sequences. q.e.d.

The tangent and the obstruction to deformation of 𝔐(ℨ^{rel}, Γ) is similar. We let S → 𝔐(Z[n]^{rel}, Γ)^{st} be an affine étale chart, satisfying the similar property as in the case just studied. We let f : X → Z[n] be the universal family with over S with D ⊂ X be the divisor of the union of all ordinary and the distinguished marked sections. We let (z_{α,1}, z_{α,2}, s_{α}) be the parameterization of charts U_{α}/V_{α} of f as in the previous case. We let Λ_{l} be those such that U_{α} ∩ f^{−1}(B_{l}) ≠ ∅. Let E^{•} and D^{•} be the complexes constructed in the Section 1 that is part of the perfect obstruction theory of S.

Proposition 5.2. The two exact sequences in Proposition 5.1 still hold with the sheaf Ω_{W [n]^{†}/A^{n+1†}} replaced by Ω_{Z[n]^{†}/A^{n†}}. The same vanishing results hold as well.

We will close this section by working out the obstruction sheaf of an example suggested to us by E. Ionel.

Let Z_{rel} = (Z, D) be a pair of smooth variety and a smooth divisor. We let Γ be the graph consisting of one vertex and one leg. We assign the weights of the vertex to be g = 1 and d = 0. Thus 𝔐(ℨ^{rel}, Γ) is the moduli of relative stable morphisms to Y from 1-pointed genus 1 curves to Z of degree 0. Since d = 0, all f: X → Z in 𝔐(ℨ^{rel}, Γ) are constant maps. Hence 𝔐(ℨ^{rel}, Γ) is isomorphic to 𝔐_{1,1} × Z. We now show that its obstruction sheaf is

Ob = π_{2}^{∗}Ω_{Z}(log D)^{∨}, (5.2)

where π_{2} :𝔐_{1,1} × Z → Z is the second projection.

Let f_{0} ∈ 𝔐(ℨ^{rel}, Γ) be a relative stable morphism. Since d = 0, we can always represent f_{0} by a morphism f_{0} : X → Z[1]^{o}, where Z[1]^{o} = Z[1] − D[1] ∪ Z[1]_{0,sing} with Z[1]_{0,sing} is the singular locus of Z[1]_{0}. Then Z[1]^{o}/ℂ^{∗} ≡ Z. The obstruction to deforming f_{0} as morphism to Z[1]^{o} is H^{1}(f_{0}^{∗}T_{Z[1]^{o}/A^{1}}) ≡ T_{Z[1]^{o}/A^{1}}|_{f_{0}(X)}, where Z[1]^{o} → A^{1} is the tautological projection and T_{Z[1]^{o}/A^{1}} is the relative tangent bundle. We let f:X → Z[1]^{o} be the family over 𝔐_{1,1}×Z[1]^{o} so that X is the pullback of the universal family over 𝔐_{1,1} while the morphism f is the composite of the projection X → 𝔐_{1,1} × Z[1]^{o} with the second projection 𝔐_{1,1} × Z[1]^{o} → Z[1]^{o}. Clearly, f is the universal family of 𝔐(Z[1]^{rel}, Γ)^{st}. The obstruction bundle to the moduli space 𝔐(Z[1]^{rel}, Γ)^{st} over 𝔐_{1,1} × Z[1]^{o} is p_{2}^{∗}T_{Z[1]^{o}/A^{1}}, where p_{2} is the second projection of 𝔐_{1,1} × Z[1]^{o}. The ℂ^{∗}-action lifts canonically to p_{2}^{∗}T_{Z[1]^{o}/A^{1}} and the obstruction sheaf of 𝔐(ℨ^{rel}, Γ) is the descent of p_{2}^{∗}T_{Z[1]^{o}/A^{1}}. It is direct to check that under the quotient map 𝔐_{1,1} × Z[1]^{o}/ℂ^{∗} ≅ 𝔐_{1,1} × Z, equivariant part (p_{2}^{∗}T_{Z[1]^{o}/A^{1}})^{ℂ^{∗}} is canonically isomorphic to π_{2}^{∗}Ω_{Z}(log D)^{∨}. This proves the identity (5.2).

5.2 Local and formal predeformable morphisms

In this part we prove that formal predeformable morphisms are automatically local predeformable.

Lemma 5.3. Let L̂ = (𝕜[z_{1}, z_{2}] ⊗_{𝕜[s]} A)ˆ , be as in Definition 1.2. Suppose there are units f_{1}, f_{2}, g_{1}, g_{2} ∈ L̂ and an integer n ≥ 1 so that f_{1}f_{2} and g_{1}g_{2} ∈ Â and that z_{i}^{m}f_{i} = z_{i}^{m}g_{i} for i = 1, 2. Then f_{1} = g_{1} and f_{2} = g_{2}.

Proof. We let r_{i} = f_{i}/g_{i}. Then z_{i}^{m}(r_{i}−1) = 0 for i = 1 and 2. Recall that elements in L̂ have unique normal form a_{0} + ∑_{i≥1}(a_{i}z_{1}^{i} + b_{i}z_{2}^{i}) for a_{i}, b_{i} ∈ Â (cf. [23]). By the uniqueness of the normal form, the normal form of r_{1} (resp. r_{2}) must be of the form r_{1} = 1 + ∑_{j≥1} a_{j}z_{2}^{j} (resp. r_{2} = 1 + ∑_{j≥1} b_{j}z_{1}^{j}). Then f_{1}f_{2}/g_{1}g_{2} = r_{1}r_{2} ∈ Â implies that 1 + ∑ a_{j}z_{2}^{j} = ϵ(1 + ∑ b_{j}z_{1}^{j})^{−1} for some unit ϵ ∈ Â, which is impossible unless all a_{j} and all b_{j} are zero. This proves the uniqueness lemma.

q.e.d.

Lemma 5.4. The notion of pure contact is independent of the choice of the charts of the nodes of U/V.

Proof. Let ϕ in (1.4) and ϕ̃: 𝕜[z̃_{1}, z̃_{2}] ⊗_{𝕜[s]} A → R be two charts of the nodes of U/V. Without loss of generality we can assume that the vanishing locus of ϕ(z_{1}) and ϕ̃(z̃_{1}) are identical in Spec R. Then by the proof of [23, Lemma 2.9], c_{i} = z_{i}/z̃_{i} ∈ R_{S} and hence c_{1}c_{2} ∈ A_{T}. Now let φ: 𝕜[w_{1}, w_{2}] → R be of pure contact with respect to the chart ϕ. If φ(w_{i}) = z_{i}^{m}h_{i} in R_{S} for h_{1}, h_{2} ∈ R_{S} satisfying h_{1}h_{2} ∈ A_{T}, then φ(w_{i}) = z_{i}^{m}(c_{i}^{m}h_{i}) in R_{S} and c_{1}^{m}h_{1}c_{2}^{m}h_{2} ∈ A_{T} as well. Thus φ is of pure contact with respect to ϕ̃ as well. This proves the lemma. q.e.d.

We now state and prove the following equivalence result.

Lemma 5.5. Let the notation be as in Definition 1.4. Then φ is of pure contact if and only if it is formally of pure contact.

Proof. Clearly, if φ is of pure m-contact, then it is so formally. We now prove the other direction. Assume φ is formally of pure m-contact. Then there are β_{1} and β_{2} ∈ R̂ such that φ̂(w_{i}) = z_{i}^{m}β_{i}, where as usual φ̂: 𝕜[w_{1}, w_{2}] → R̂ is the homomorphism induced by φ. We first show that there are f_{1} and f_{2} ∈ R_{S} so that φ(w_{i}) = z_{i}^{m}f_{i} in R_{S} for i = 1 and 2. Let x = φ(w_{1}) ∈ R, then x̂ = φ̂(w_{1}) ∈ z_{1}^{m}R̂. Thus by [29, Thm 8.1], the residue class of \overline{x} ∈ R_{S}/(z_{1}^{m}) is (\overline{x})ˆ = 0 ∈ (R/(z_{1}^{m}))ˆ . Hence \overline{x} ∈ ∩_{m≥1}I^{m}R/(z_{1}^{m}), and by [29, Thm 8.9] there is an a ∈ R/(z_{1}^{m}) satisfying a ≡ 1 mod I such that a\overline{x} = 0. Then a ∈ S and by our assumption a is a unit in R_{S}/(z_{1}^{m}). Hence \overline{x} = 0 in R_{S}/(z_{1}^{m}). This proves that φ(w_{1}) ∈ z_{1}^{m}R_{S} and hence there is an f_{1} ∈ R_{S} such that φ(w_{1}) = z_{1}^{m}f_{1} in R_{S}. For the same reason, φ(w_{2}) = z_{2}^{m}f_{2} in R_{S} for some f_{2} ∈ R_{S}.

We next prove the following induction hypothesis: For any nonnegative integer k, there are g_{1} and g_{2} ∈ R_{S} such that

z_{1}^{m}f_{1} − z_{1}^{m}g_{1}, z_{2}^{m}f_{2} − z_{2}^{m}g_{2} ∈ s^{k}R_{S} and g_{1}g_{2} ∈ A_{T} + s^{k}R_{S}. (5.3)

Clearly, this statement is true for k = 0. We now show that this statement is true for k + 1 if it is true for k. Let g_{1} and g_{2} be elements in R_{S} satisfying (5.3) for an integer k. To carry out the induction we need to find r_{1} and r_{2} ∈ R_{S} so that

z_{i}^{m}f_{i} − z_{i}^{m}(g_{i} + r_{i}) ∈ s^{k+1}R_{S} for i = 1, 2 (5.4)

and

(g_{1} + r_{1})(g_{2} + r_{2}) ∈ A_{T} + s^{k+1}R_{S}. (5.5)

Since φ formally is of pure m-contact, there are units η_{1} and η_{2} ∈ R̂_{S} such that z_{i}^{m}f̂_{i} = z_{i}^{m}η_{i} for i = 1 and 2 and η_{1}η_{2} ∈ Â. Because ĝ_{i} and η_{i} satisfy the relation

z_{i}^{m}ĝ_{i} = z_{i}^{m}η_{i} mod s^{k} and ĝ_{1}ĝ_{2}, η_{1}η_{2} ∈ Â + s^{k}R̂,

by the uniqueness Lemma 5.3, ĝ_{i} ≡ η_{i} mod s^{k}. Hence

(z_{i}^{m}f_{i} − z_{i}^{m}g_{i})ˆ = z_{i}^{m}f̂_{i} − z_{i}^{m}ĝ_{i} = z_{i}^{m}(η_{i} − ĝ_{i}) ∈ z_{i}^{m}s^{k}R̂.

As we argued in the existence of f_{i}, this implies that z_{i}^{m}(f_{i} − g_{i}) ∈ z_{i}^{m}s^{k}R_{S}.

We now let l_{i} ∈ R_{S} be such that z_{i}^{m}f_{i} − z_{i}^{m}g_{i} = z_{i}^{m}s^{k}l_{i}. We let r_{1} = s^{k}(l_{1} + z_{2}h_{1}) and r_{2} = s^{k}(l_{2} + z_{1}h_{2}) with h_{1} and h_{2} ∈ R_{S} to be determined. Clearly, (5.4) will hold with this choice of r_{1} and r_{2}. As for (5.5), we need

(g_{1}+r_{1})(g_{2}+r_{2}) = g_{1}g_{2}+s^{k}(g_{1}l_{2}+g_{1}z_{1}h_{1}+g_{2}l_{1}+g_{2}z_{2}h_{2}) ∈ A_{T} +s^{k+1}R_{S}.

Since g_{1}g_{2} ∈ A_{T} + s^{k}R_{S}, we can find c ∈ R_{S} so that g_{1}g_{2} − cs^{k} ∈ A_{T} + s^{k+1}R_{S}. Hence we need to find h_{1} and h_{2} in R_{S} so that

(c + g_{1}l_{2} + g_{2}l_{1}) + g_{1}h_{1}z_{1} + g_{2}h_{2}z_{2} ∈ A_{T} + sR_{S}.

Let d = c + g_{1}l_{2} + g_{2}l_{1}. Clearly, there are γ_{1} and γ_{2} ∈ R̂ and α ∈ A_{T} so that d̂− (α + γ_{1}z_{1} + γ_{2}z_{2}) ∈ sR_{S}. Hence following the argument for the existence of f_{i}, there are c_{1} and c_{2} ∈ R_{S} so that d − (c_{1}z_{1} + c_{2}z_{2}) ∈ A_{T} + sR_{S}. Hence the choice h_{1} = −c_{1}g_{1}^{−1} and h_{2} = −c_{2}g_{2}^{−1} will do the job for (5.5). Here g_{1} and g_{2} are units since β_{1} and β_{2} are units. This proves that for each k we can find g_{1} and g_{2} that satisfy (5.3).

Now we show that there are h_{1} and h_{2} ∈ R_{S} as required by the lemma. We first let M ⊂ R_{S} be the set of those elements that are annihilated by some power of s. It is an ideal, and since R_{S} is Noetherian, there is an N so that s^{N}M = 0. We let g_{1} and g_{2} be the pair satisfying (5.3) for k = N + 1. In case R/M = 0 then the lemma is already proved. Now assume R/M ≠ 0. We consider the ring R_{S}/M_{S}. Since φ̂(t) = φ̂(w_{1})φ̂(w_{2}) = ϵs^{m} in R̂ for some unit ϵ ∈ R̂, R_{S}/M_{S} is flat over 𝕜[t]. Now consider the homomorphism \overline{φ} : 𝕜[w_{1}, w_{2}] ⟶ R_{S}/M_{S} induced by φ. By [23, Prop. 2.2], it is formally of pure contact. Hence there are \overline{h}_{1} and \overline{h}_{2} ∈ R_{S}/M_{S} so that \overline{φ}(w_{i}) = z_{i}^{m}\overline{h}_{i} in R_{S}/M_{S} for i = 1 and 2. Because R_{S}/M_{S} is flat over 𝕜[t], \overline{φ}_{1}(w_{1})\overline{φ}_{2}(w_{2}) ∈ R_{S}/M_{S} implies that \overline{h}_{1}\overline{h}_{2} ∈ R_{S}/M_{S}. We then apply the uniqueness Lemma 5.3 to conclude that the residue classes of \overline{h}_{i} and g_{i} in R_{S}/(M_{S}, s^{k}) are identical. Hence we can find h_{i} ∈ R_{S} so that its residue class in R_{S}/M_{S} and R_{S}/(s^{k}) are \overline{h}_{i} and g_{i} respectively. Therefore we have φ(w_{i}) = z_{i}^{m}h_{i} in R_{S} and h_{1}h_{2} ∈ A_{T}. This completes the proof of the lemma. q.e.d.

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