j. differential geometry 66 (2004) 303-343

HITCHIN’S CONNECTION AND DIFFERENTIAL OPERATORS WITH VALUES IN THE DETERMINANT

BUNDLE

Xiaotao Sun & I-Hsun Tsai

Abstract

For a family of smooth curves, we have the associated family of moduli spaces of stable bundles with fixed determinant on the curves. There exists a so-called theta line bundle on the family of moduli spaces. When the Kodaira–Spencer map of the family of curves is an isomorphism, we prove in this paper an identification theorem between sheaves of differential operators on the theta line bundle and higher direct images of vector bundles on curves. As an application, the so-called Hitchin connection on the direct image of (powers of) the theta line bundle is derived naturally from the identification theorem. A logarithmic extension to certain singular stable curves is also presented in this paper.

1. Introduction

Let M be a smooth variety over any algebraically closed field of characteristic zero and S → M be a smooth, flat morphism. For a smooth family π : X → S of curves and a vector bundle E on X, A. Beilinson and V. Schechtman defined in [2] (see also [4]) the so-called trace complexes ^{tr} A_{E/M}^{•}, ^{tr} A_{E}^{•} on X, together with algebra structures on them, such that R^{0}π_{∗}(^{tr} A_{E/M}^{•}), R^{0}π_{∗}(^{tr} A_{E}^{•}) are canonically isomorphic to the Atiyah algebras A_{λ_{E}/M}, A_{λ_{E}} of the determinant bundle λ_{E} = det Rπ_{∗}E of the family (see Proposition 3.3 for details), where A_{λ_{E}}, as a sheaf on S, is just D_{S}^{≤1}(λ_{E}), the sheaf of differential operators of order ≤ 1 with values in λ_{E}, and similarly A_{λ_{E}/M} ⊂ A_{λ_{E}} corresponds to relative differential operators. It is generalized by Y.-L. Tong and the second author ([19]) to the families π : X̃ → S̃ of stable curves. In this paper, we prove firstly, under the assumption that Kodaira–Spencer maps of the families are isomorphisms, one can canonically produce locally free sheaves G_{E}, S(G_{E}) on X (resp. on X̃) instead of the above trace complexes such that one has the following identification theorem:

Sun is supported by the outstanding young grant of NFSC at number 10025103 and a grant of RGC of Hong Kong at RGC code 7131/02P. Tsai has been supported in part by a grant from National Science Council of Taiwan.

Received 11/18/2003.

Theorem 1.1.

i) If E is fiberwise stable and the Kodaira–Spencer map

KS : T_{S/M} → R^{1}π_{∗}(𝔼)

is an isomorphism, where 𝔼 := End^{0}(E) is the trace free part of End(E), then there exists a canonical isomorphism

ϕ : R^{1}π_{∗}(G_{E}) ≅ D_{S/M}^{≤1}(λ_{𝔼}).

ii) If KS_{S} : T_{S} → R^{1}π_{∗}A_{E/S}^{0} is an isomorphism, then (after shrinking M enough) ϕ extends to an isomorphism (cf. Theorem 2.5 for details)

ϕ = ϕ_{−λ} : R^{1}π_{∗}S(G_{E}) ≅ D_{S}^{≤1}(λ_{𝔼}).

The most technical part of this paper is devoted to proof of the identification theorem and its generalization to stable curves π : X̃ → S̃ (cf. Theorem 2.13). There is an interesting example of the families S → M, π : X → S which satisfy the assumptions in our theorem. Let C → M be a family of smooth curves of genus g ≥ 2 and f : S → M be the associated family of moduli spaces of stable bundles of rank r ≥ 2 with a fixed determinant. Let π : X = C ×_{M} S → S, E a universal bundle on X. Then KS : T_{S/M} → R^{1}π_{∗}(𝔼) is an isomorphism. If furthermore, for any b ∈ M, T_{M,b} ≅ H^{1}(C_{b}, T_{C_{b}}), then KS_{S} : T_{S} → R^{1}π_{∗}A_{E/S}^{0} is an isomorphism (cf. Propsition 4.5). A family C → M as such (called good family for convenience) does exist in algebraic category of characteristic zero (in fact, M is a moduli space of curves with level structures, which is a finite Galois cover of the moduli space M_{g} of curves). We will fix such a family in (the most part of) this paper. Let Θ be the theta line bundle on S and, for any k, write Θ^{k} = λ_{𝔼}^{μ} for some μ ∈ ℚ. Then our identification theorem gives immediately a new construction of Hitchin’s connection on V_{k} = f_{∗}(Θ^{k}).

Corollary 1.2. There exists a unique projective heat operator,

H : T_{M} → f_{∗}D_{S}^{≤2}(Θ^{k})/O_{M}

which lifts Hitchin’s symbol map f_{∗}δ_{H} : T_{M} → f_{∗}S^{2}T_{S/M}.

The construction of the above heat operator is purely algebraic by playing diagram game. The map δ_{H} : f^{∗}T_{M} → S^{2}T_{S/M} is a connecting map in a long exact sequence of cohomologies, which coincides with Hitchin’s symbol map over complex number field. The reason we restrict ourselves in characteristic zero is: the theory of Beilinson–Schechtman (thus our identification theorem) is in characteristic zero. On the other hand, Tsuchiya–Ueno–Yamada [20] construct a conformal block space associated to each pointed-stable curve, these spaces form a vector bundle on the moduli space of stable curves. They also construct a logarithmic projective connection (WZW connection) on this vector bundle of conformal blocks. It is proved that the algebro-geometric analogues of conformal block spaces for smooth curves are the spaces of global sections of Θ^{k}. Y. Laszlo [13] proves that on the moduli space M_{g,1} of smooth curves with a point Hitchin’s connection coincides with the WZW connection (note that M_{g,1} is of dimension larger than our M). However, for singular stable curves, the algebro-geometric analogues of conformal block spaces are still unknown (by our knowledge). One reason, among others, is simply that degeneration of moduli spaces of stable SL(r)-bundles is still an open problem (cf. [6], [7] for comments on this problem). Thus it seems to us that a geometrically meaningful extension of V_{k} = f_{∗}(Θ^{k}) to the boundary with logarithmic connection would be interesting. Indeed, the question of studying the behaviour of spaces of generalized theta functions under degeneration of the curves was already mentioned in [10], p. 350, and the question of studying the (logarithmic) extension of the connection was asked in an unpublished manuscript of Beilinson and Kazhdan [1], p. 27, where they produced the connection for the case of smooth curves.

Let S̃, M̃ be smooth varieties and B = M̃ \ M be a normal crossing divisor. Let f̃ : S̃ → M̃ be a smooth flat morphism, π : X̃ → S̃ be a family of stable curves and E a fiberwise stable vector bundle on X̃. Let W = f̃^{−1}(B) ⊂ S̃. Then the logarithmic identification theorem can be proved similarly as in the smooth case.

Theorem 1.3.

i) If KS : T_{S̃/M̃} →R^{1}π_{∗}(𝔼) is an isomorphism, where 𝔼:= End^{0}(E), then there exists a canonical isomorphism

ϕ : R^{1}π_{∗}(G_{E}) ≅ D_{S̃/M̃}^{≤1}(λ_{𝔼}).

ii) If KSS̃ : TS̃(log W) → R^{1}π_{∗}A_{E/S̃}^{0}(log D) is an isomorphism, then

(after shrinking M̃ enough) ϕ extends to an isomorphism (cf. Theorem 2.13 for details)

ϕ = ϕ_{−λ} : R^{1}π_{∗}S(G_{E}) ≅ D≤1 ; S̃(λ_{𝔼})(log W).

If C̃ → M̃ is a family of stable curves, let U(r, d) → M̃ be the family of moduli spaces of semistable (for canonical polarization) torsion-free sheaves of rank r and degree d. Fix a line bundle N on C̃ of relative degree d, let f : S → M be the family of moduli spaces of stable bundles of rank r with fixed determinant N |_{C} ([C] ∈ M). Let Z ⊂ U(r, d) be the Zariski closure of S in U(r, d) and f_{Z} : Z → M̃ the induced morphism. Let f̃: S̃ ⊂ Z → M̃ be the open set of Z consisting of stable bundles. Let π : X̃ = C̃×_{M} S̃ → S̃ and E the universal bundle on X̃. Note that for general reducible stable curves C̃_{b} (b ∈ B), the (open) moduli spaces f̃^{−1}(b) hasn’t been well understood yet, and in particular we do not know if the Kodaira–Spencer map in Theorem 1.3 i) is an isomorphism. On the other hand it is better understood for those irreducible stable curves and reducible stable curves whose dual diagrams are trees. Thus we will assume in this paper that the singular fibers of C̃ → M̃ are (irreducible or reducible) stable curves with only one node. Then f̃: S̃ → M̃ is smooth and the Kodaira–Spencer map in Theorem 1.3 i) is an isomorphism (cf. Lemma 4.4). Let M̃_{g} be the moduli space of stable curves with level structures as in Looijenga’s paper [14], with the universal family Π : C̃ → M̃_{g}. On a neighbourhood of m ∈ M̃_{g}, the family is the versal deformation of C̃_{m}. In particular, KS : TM̃(B) → R^{1}Π_{∗}ω_{C̃/M̃}^{−1} is an isomorphism, which implies that the Kodaira–Spencer

map in Theorem 1.3 ii) is an isomorphism (cf. Proposition 4.5). We will fix M̃ ⊂ M̃_{g} consisting of stable curves with at most one double point and also fix the family C̃ → M̃ in this paper. It is known that B = M̃ \ M is a normal crossing divisor (cf. [14]).

Corollary 1.4. There exists a unique projective heat operator

H̃ : TM̃(B) → f̃_{∗}D≤2 ; S̃(Θ^{k})/OM̃

that lifts Hitchin’s symbol map (in the extended sense)

f̃_{∗}δ_{H} : TM̃(B) → f̃_{∗}S^{2}T_{S̃/M̃}.

Moreover, f̃_{∗}(Θ^{k}) is a coherent sheaf on M̃.

After defining the extended Hitchin’s symbol map, the proof goes through by using the logarithmic version of identification theorem. It is important that the vanishing of R^{1}f_{∗}O_{S} is not needed in our proof both for smooth case and singular case (so that our method can be extended to some stable curves, where such a vanishing may not hold). The nontrivial part is the proof of coherence of f̃_{∗}(Θ^{k}) on M since f̃: S̃ → M̃ is not proper. To prove it, we prove (cf. Proposition 4.9) that the fibres of f̃ : S̃ → M̃ is dense in the fibres of f_{Z} : Z → M̃, which partially confirms a conjecture of D. S. Nagaraj and C. S. Seshadri (cf. Conjecture (a) at page 136 of [15]). We believe that such a coherent extension remains true on M̃_{g} of all stable curves; this is not, however, proved in the present paper. On the other hand, if it can be proved that the compactification Z is Cohen–Macaulay with vanishing H^{1}(f_{Z}^{−1}(b), Θ^{k}) (b ∈ B), then the coherent extension is a vector bundle whose fibers are spaces of generalized theta functions on certain moduli spaces. This seems to be true for rank two case.

Basically our approach works equally well both for the general case [10] (g ≥ 3) and special cases (e.g., g = 2 and r = 2) [8], with the difference arising from the fact that in g = r = 2 case our approach shall work with a Quot scheme R^{ss} which has a good quotient R^{ss} → S where S is the family of moduli spaces of S-equivalence classes of semistable vector bundles. As this will introduce additional details, we leave the case g = r = 2 to future discussions.

Hitchin remarked in [10], p. 350, “Many aspects of the connection have so far resisted attack by the methods presented here...”. Besides the degeneration problem treated here, it seems that two major problems remain poorly understood:

i) the unitary structure of the connection, ii) (local) monodromy representation of the connection,

although there are discussions on these problems in the literature. For ii) we refer to [8] for interesting results for g = r = 2 case along the line of Hitchin’s approach; for comments on more general cases, see [8], Subsection 4.5 and references therein. We content ourselves by remarking that the methods for studying the monodromy representation (of the mapping class group) have largely been confined to those of topology and conformal field theory. An algebro-geometric picture is still lacking for the understanding (note that [8] merely found local monodromy representation; “local” means at ∞). In the spirit of the work [10], [13] and [8], we feel it worthy of being presented, among other things, the present proof of the existence of the logarithmic Hitchin’s connection (beyond the explicit g = r = 2 case of [8]), as a further step towards algebro-geometric aspects of these development. In our opinion it seems plausible that some explicit information about the (local) monodromy representation could be available via our approach. Indeed in the first version of this paper we produced a more explicit formula of the connection, at the expense that a lot more technical computations would be performed, while the present approach simplified considerably via use of Proposition 2.3 which we later discovered to be useful (in fact we came to pay more attention to this proposition via discussions with Hélène Esnault when we finished the first version including the case g = r = 2). For the sake of conceptual simplicity we decided to take the present approach, and leave the more technical treatment (including the case g = r = 2) to future discussions. We remark that there has been a speculation about whether monodromy representations factor through a finite group; see [3] where this speculation was attributed to M. Kontsevich. We had also met with mathematicians who asked about (and/or suggested the study of) the characteristic p > 0 situation, including the related issues such as the notion of p-curvature of the connection, in connection to the highly speculative relation between the above Kontsevich conjecture and some conjectures of Grothendieck (cf. [12]). Unfortunately at this time of writing we are not feeling competent enough to comment on any one of these issues, based on results of the present work. Perhaps some further studies along these directions might be interesting. Besides generalizing Beilinson–Schechtman theory (thus our construction) to characteristic p > 0, the logarithmic extension suggests that residue of the connection at the boundary gives an operator on the (projective) space of generalized theta functions on certain moduli space, which should be (direct sum of) spaces of generalized theta functions on moduli spaces of parabolic bundles over a smooth curve of lower genus (normalization of the singular curve C̃_{b}). It would be interesting to understand the relationship between the residue operator and the above factorization of generalized theta functions.

Those who want to know some historical accounts of Hitchin’s connection and its relation to other branches of mathematics, are referred to [10], [8] and references therein.

In Section 2, we construct Hitchin’s connection and its logarithmic extension under assumption of Theorem 2.5, Lemma 2.4 and Theorem 2.13. It is enough to read this section for a reader who is only interested in Hitchin’s connection. Section 3 is devoted to the proof of Theorem 2.5 and Lemma 2.4. In Section 4, we indicate the modifications of arguments in Section 3 to prove Theorem 2.13. Most of Section 4 is devoted to prove the coherence of f̃_{∗}(Θ^{k}) on M̃.

Remark that some ideas closely related to Section 3 of this paper were briefly discussed in Sections 9, 10 of [9] where the subject matter is considered for moduli space of (stable) G-bundles, with G being complex semisimple, connected and simply-connected.

Acknowledgements. We would like to express our deep gratitude to Hélène Esnault for the great impact of her ideas on this work. Indeed countless discussions with her in an earlier stage of this work either directly allowed us to grasp several key ideas, or contributed in an essential way to shaping ourselves towards the present approach. This joint work was initiated while both of us were visiting Essen (though a few topics involved in this work had already been studied by three of us independently). We wish to thank heartily Hélène Esnault and Eckart Viehweg for their warm hospitality and the excellent working conditions of Essen, that made our visit and this work possible. The anonymous referees have given us a critical analysis of several related topics that would be worth further pursuit; their thorough reading of this paper and supply of some relevant information have led to substantial improvement on our presentation. We thank them very much.

2. Heat operators and cohomology of sheaves

As in the introduction, M ⊂ M̃_{g} denote the smooth moduli spaces of curves with level structures ([14]). Let M̃ ⊂ M̃_{g} be the open set consisting of stable curves with at most one double point (thus codim (M̃_{g} \ M̃) ≥ 2). We will fix the families of curves C → M, C̃ → M̃ which satisfy the assumptions on Kodaira–Spencer maps. Fix a line bundle N of relative degree d on C̃ → M̃, let f : S → M be the family of moduli spaces of stable bundles of rank r with fixed determinant N_{b} := N |_{C_{b}} (b ∈ M) on C_{b}. Let π : X = C ×_{M} S → S be the pullback of C → M via f : S → M.

Remark 2.1. If (d, r) = 1, there exists a universal bundle E on X. In general S is a good quotient of a Hilbert quotient scheme, denoted by R^{s}, such that there is a universal bundle E on X_{R^{s}} := C ×_{M} R^{s}. E may not descend to X, but objects such as End^{0}(E), G_{E}, ^{tr} A_{E}^{•} (and other relevant constructions that will be discussed later in this section) do descend. Recall that a sheaf F on X_{R^{s}} descends to X if the action, of scalar automorphisms of E (relative to R^{s}), on F is trivial, e.g., [11].

Without the danger of confusion we will henceforth be working as if a universal bundle E existed on X.

Let Θ be the theta line bundle on S. For a sheaf F on X,

λ_{F} = ⨂_{q≥0} det R^{q}π_{∗}F^{(−1)^{q}}

denotes the Knudsen–Mumford determinant bundle on S (cf. [2]). As usual T_{S}, K_{S} denote tangent bundle, canonical bundle; T_{S/M}, K_{S/M} denote the relative counterparts. Assume that K_{S/M} = Θ^{−λ}, where λ = 2(r, d) is the same notation used by Hitchin (cf. (2.8) in [10]). Let D_{S}^{≤i}(Θ^{k}) be the sheaf of differential operators on Θ^{k} of order ≤ i, and by ϵ the symbol map of differential operators. Let W(Θ^{k}) := D_{S}^{≤1}(Θ^{k})+ D_{S/M}^{≤2}(Θ^{k}) (cf. [10], [8]), one has an exact sequence

0 → D_{S/M}^{≤2}(Θ^{k}) → W(Θ^{k}) ⟶^{σ} f^{∗}T_{M} → 0. (2.1)

Definition 2.2 (cf. [8], 2.3.2). A heat operator H on Θ^{k} is an O_{S}- map f^{∗}T_{M} ⟶^{H} W(Θ^{k}) which, while composed with σ above, is the identity. A projective heat operator H : T_{M} → f_{∗}W(Θ^{k})/O_{M} is an O_{M}- map such that any local lifting is a heat operator. The symbol map of H is ϵ ∘ H : f^{∗}T_{M} → S^{2}T_{S/M}. A heat operator H induces an O_{M}-map (denoted by the same H) H: T_{M} → f_{∗}W(Θ^{k}). The heat operator and the preceding induced map will be used interchangeably throughout. A (projective) heat operator on Θ^{k} determines a (projective) connection ∇^{H} on f_{∗}Θ^{k} in a natural way [8].

We recall firstly a general result (forget moduli) of G. Faltings (cf. Proposition IV.5 of [5]). Let Z → S be smooth and K = K_{Z/S}. For any line bundle L, consider

0 → D_{Z/S}^{≤1}(L) → D_{Z/S}^{≤2}(L) ⟶^{ϵ_{2}} S^{2}T_{Z/S} → 0

0 → D_{Z/S}^{≤1}(L) → S^{2}D_{Z/S}^{≤1}(L) ⟶^{S^{2}ϵ_{1}} S^{2}T_{Z/S} → 0.

For any ρ ∈ H^{0}(Z, S^{2}T_{Z/S}), let a(L, ρ), b(L, ρ) be the obstruction classes to lift ρ to H^{0}(Z, D_{Z/S}^{≤2}(L)) and to H^{0}(Z, S^{2}D_{Z/S}^{≤1}(L)) respectively. Then:

Proposition 2.3. Under D_{Z/S}^{≤1}(L^{k}) ≅ D_{Z/S}^{≤1}(L), one has:

i) b(L^{k}, ρ) = kb(L, ρ) for any k ∈ ℚ.

ii) a(O_{Z}, ρ) ∈ H^{1}(O_{Z}) ⊕ H^{1}(T_{Z/S}) has zero projection in H^{1}(O_{Z}). iii) There is a class c(K, ρ) ∈ H^{1}(O_{Z}), independent of L, such that

2a(L, ρ) = b(L, ρ) + ^{t}b(L^{−1} ⊗ K, ρ) + c(K, ρ).

We come back to construct the projective connection. Recall that

X = C ×_{M} S ⟶^{π} S
p↓  ^{f}↓
C  ⟶ M.

For simplicity, we assume that there exists a universal bundle E on X, let 𝔼 = End^{0}(E). We will arrive at a subsheaf G_{E} ⊂ ^{tr} A_{E}^{−1} (see [2] for details of ^{tr} A_{E}^{−1}) fitting into an exact sequence

0 → ω_{X/S} → G_{E} ⟶^{res} 𝔼 → 0, (2.2)

which induces, by taking the 2-nd symmetric tensor, the exact sequence

0 → G_{E} → S^{2}(G_{E}) ⊗ T_{X/S} ⟶^{Sym^{2}(res)⊗id} S^{2}(𝔼) ⊗ T_{X/S} → 0.

Define S(G_{E}) := (Sym^{2}(res) ⊗ id)^{−1}(id ⊗ T_{X/S}), which fits into

0 → G_{E} ⟶^{ι} S(G_{E}) ⟶^{q} T_{X/S} → 0, (2.3)

where q = Sym^{2}(res) ⊗ id, ι(α) = Sym^{2}(α ⊗ dt) ⊗ ∂_{t} locally. Let

0 → R^{1}π_{∗}(ω_{X/S}) → R^{1}π_{∗}(G_{E}) ⟶^{res} R^{1}π_{∗}(𝔼) → 0 (2.4)

0 → R^{1}π_{∗}(G_{E}) ⟶^{ι} R^{1}π_{∗}S(G_{E}) ⟶^{q} R^{1}π_{∗}(T_{X/S}) → 0 (2.5)

be the exact sequences induced by (2.2), (2.3). Let Δ ⊂ X ×_{S} X be the diagonal, consider the induced diagram

0 ⟶ G_{E} ⊠ G_{E} ⟶ G_{E} ⊠ G_{E}(Δ) ⟶ G_{E} ⊠ G_{E}(Δ)|_{Δ} ⟶ 0
↓  ↓  ↓
0 ⟶ 𝔼 ⊠ 𝔼 ⟶ 𝔼 ⊠ 𝔼(Δ) ⟶ 𝔼 ⊠ 𝔼(Δ)|_{Δ} ⟶ 0

on X ×_{S} X (𝔼⊠𝔼 denotes p_{1}^{∗}𝔼⊗p_{2}^{∗}𝔼). All vertical maps are induced by G_{E} ⟶^{res} 𝔼 → 0, thus (2.3) is a subsequence of the rightmost vertical map above. Let F_{1} ⊂ G_{E} ⊠ G_{E}(Δ), F_{2} ⊂ 𝔼 ⊠ 𝔼(Δ) be subsheaves satisfying

0 ⟶ G_{E} ⊠ G_{E} ⟶ F_{1} ⟶ S(G_{E}) ⟶ 0
↓  ↓  ^{q}↓
0 ⟶ 𝔼 ⊠ 𝔼 ⟶ F_{2} ⟶ T_{X/S} ⟶ 0.

Taking direct images, considering the connecting maps, we have

R^{1}π_{∗}S(G_{E}) \underrightarrow{δ̃} S^{2}R^{1}π_{∗}(G_{E})
q↓  ^{S^{2}(res)}↓
R^{1}π_{∗}(T_{X/S}) ⟶^{δ} S^{2}R^{1}π_{∗}(𝔼)

which induces the commutative diagram

R^{1}π_{∗}(G_{E}) ⟶^{δ_{1}} R^{1}π_{∗}(G_{E})
ι↓  ^{Sym}^{2}↓
R^{1}π_{∗}S(G_{E}) \underrightarrow{δ̃} S^{2}R^{1}π_{∗}(G_{E})
q↓  ^{S^{2}(res )}↓
R^{1}π_{∗}(T_{X/S}) ⟶^{δ} S^{2}R^{1}π_{∗}(𝔼)

where δ_{1} is defined such that the diagram is commutative, the first vertical exact sequence is (2.5), the second vertical exact sequence is induced by taking the 2-nd symmetric tensor of (2.4) (note that O_{S} ≅ R^{1}π_{∗}(ω_{X/S})). For Sym^{2} see Remark 2.6.

Lemma 2.4. The map δ_{1} : R^{1}π_{∗}(G_{E}) → R^{1}π_{∗}(G_{E}) is the identity map.

Proof. See Lemma 3.18. q.e.d.

Theorem 2.5.

i) There is an isomorhism ϕ : R^{1}π_{∗}(G_{E}) ≅ D_{S/M}^{≤1}(λ_{𝔼}) such that the following diagram is commutative:

0 ⟶ O_{S} =R^{1}π_{∗}ω_{X/S} ⟶ R^{1}π_{∗}(G_{E}) ⟶^{res} R^{1}π_{∗}(𝔼) ⟶ 0
(2r)·id↓  ^{ϕ}↓  _{ϑ}↓
0 ⟶ O_{S} =R^{0}π_{∗}ω_{X/S}[1] ⟶ D_{S/M}^{≤1}(λ_{𝔼}) ⟶^{ϵ_{1}} T_{S/M} ⟶ 0

where ϑ is a Kodaira–Spencer type identification. ii) For any affine covering {M_{i}}_{i∈I} of M, on each S_{i} := f^{−1}(M_{i}),

there is an isomorphism ϕ_{i} : R^{1}π_{∗}S(G_{E}) ≅ D_{S}^{≤1}(λ_{𝔼}) such that

0 ⟶ R^{1}π_{∗}(G_{E}) ⟶^{2·ι} R^{1}π_{∗}S(G_{E}) ⟶^{q} R^{1}π_{∗}(T_{X/S}) ⟶ 0
ϕ↓  ^{ϕ_{i}}↓  ↓
0 ⟶ D_{S/M}^{≤1}(λ_{𝔼}) ⟶ D_{S}^{≤1}(λ_{𝔼}) ⟶^{σ} f^{∗}T_{M} ⟶ 0

is commutative on S_{i}. Moreover, {ϕ_{i} − ϕ_{j}} defines a class in H^{1}(Ω_{M}^{1}).

Proof. See Theorem 3.7, Corollary 3.12 and Remark 3.17. q.e.d.

By the above theorem, on each S_{i}, we get commutative diagram

D_{S/M}^{≤1}(λ_{𝔼}) ⟶^{ϕ^{−1}} R^{1}π_{∗}(G_{E}) ⟶^{δ_{1}} R^{1}π_{∗}(G_{E}) ⟶^{2r·ϕ} D_{S/M}^{≤1}(λ_{𝔼})
↓  ^{2·ι}↓  ^{Sym}^{2}↓  ↓
D_{S}^{≤1}(λ_{𝔼}) ⟶^{ϕ_{i}^{−1}} R^{1}π_{∗}S(G_{E}) \underrightarrow{δ̃} S^{2}R^{1}π_{∗}(G_{E}) ⟶^{S^{2}(ϕ)} S^{2}D_{S/M}^{≤1}(λ_{𝔼})
σ↓  ↓  ^{q}↓  _{S^{2}ϵ_{1}}↓
f^{∗}T_{M} ⟶^{≅} R^{1}π_{∗}(T_{X/S}) ⟶^{δ} S^{2}R^{1}π_{∗}(𝔼) ⟶^{≅} S^{2}T_{S/M}.

Remark 2.6.

i) For a precise definition of Sym^{2} above we refer to proof of Lemma 3.18.

ii) The insertion of 2r in 2r · ϕ in the first row is due to Theorem 2.5 i) combined with the definition of Sym^{2}.

The above diagram gives the following commutative diagram on S_{i}:

0 ⟶ D_{S/M}^{≤1}(λ_{𝔼}) ⟶ D_{S}^{≤1}(λ_{𝔼}) ⟶^{σ} f^{∗}T_{M} ⟶ 0
∥  δ̃_{H}^{i}↓  ^{δ_{H}}↓
0 ⟶ D_{S/M}^{≤1}(λ_{𝔼}) ⟶ S^{2}D_{S/M}^{≤1}(λ_{𝔼}) ⟶^{S^{2}ϵ_{1}} S^{2}T_{S/M} ⟶ 0

where δ_{H}, δ̃_{H}^{i} are defined in a clear way such that each δ̃_{H}^{i} induces an identity map on D_{S/M}^{≤1}(λ_{𝔼}).

Proposition 2.7. Replace M by an affine open set. For any ρ = δ_{H}(v) ∈ H^{0}(S, S^{2}T_{S/M}), where 0 ≠ v ∈ H^{0}(S, f^{∗}T_{M}), one has:

i) 2a(L, ρ) = b(L, ρ) + ^{t}b(L^{−1} ⊗ K_{S/M}, ρ).

ii) When L = K_{S/M}^{μ}, where μ ∈ ℚ and μ ≠ 1, one has

a(K_{S/M}^{μ}, ρ) = \frac{2μ − 1}{2μ}b(K_{S/M}^{μ}, ρ).

Proof. To prove i), by Proposition 2.3 iii), it is enough to show that c(K, ρ) = 0. The class is independent of L. Thus, by taking L = O_{S} and using Proposition 2.3 ii), it is enough to show that

tb(K, ρ) ∈ H^{1}(D_{S/M}^{≤1}(O_{S})) = H^{1}(O_{S}) ⊕ H^{1}(T_{S/M})

has trivial projection in H^{1}(O_{S}) where K = K_{S/M}. Using Theorem 2.5 (we need here that M be replaced by an affine open set), we have (noting K = λ_{𝔼})

D_{S/M}^{≤1}(K)  D_{S/M}^{≤1}(K)
↓  ↓
D_{S}^{≤1}(K) \underrightarrow{δ̃_{H}} S^{2}D_{S/M}^{≤1}(K)
σ↓  _{S^{2}ϵ_{1}}↓
f^{∗}T_{M} ⟶^{δ_{H}} S^{2}T_{S/M}.

Let {U_{i}}_{i∈I} be an affine covering of S and v_{i} ∈ T_{S}(U_{i}) be local liftings of v ∈ f^{∗}T_{M}(S). Let {d_{i} ∈ D_{S}^{≤1}(K)(U_{i})}_{i∈I} be such that ϵ_{1}(d_{i}) = v_{i} (i ∈ I). Then, by above diagram, b(K, ρ) = {d_{i} − d_{j} ∈ D_{S/M}^{≤1}(K)(U_{i} ∩ U_{j})}.

Thus the class ^{t}b(K, ρ) ∈ H^{1}(D_{S/M}^{≤1}(O_{S})) = H^{1}(O_{S}) ⊕ H^{1}(T_{S/M}) is

defined by the cocycle {^{t}d_{i} − ^{t}d_{j}}, where ^{t}d_{i} := A_{i} −v_{i} ∈ D_{S}^{≤1}(O_{S})(U_{i}).

Thus the projection of ^{t}b(K, ρ) in H^{1}(O_{S}) is defined by {A_{i}−A_{j}}, which is a trivial class.

To show ii), we remark that for any nonzero μ ∈ ℚ, through canonical isomorphisms ψ_{μ} : D^{≤1}(K) ≅ D^{≤1}(K^{μ}), the above diagram induces

D_{S/M}^{≤1}(K^{μ}) ⟶^{•μ} D_{S/M}^{≤1}(K^{μ})
↓  ↓
D_{S}^{≤1}(K^{μ}) \underrightarrow{δ̃_{H}^{μ}} S^{2}D_{S/M}^{≤1}(K^{μ})
σ↓  _{S^{2}ϵ_{1}}↓
f^{∗}T_{M}  ⟶^{δ_{H}} S^{2}T_{S/M}

where δ̃_{H}^{μ} = S^{2}ψ_{μ} ∘ δ̃_{H} ∘ ψ_{μ}^{−1}. Using the above diagram, we can compute tb(K^{1−μ}, ρ). Let {d_{i} ∈ D_{S}^{≤1}(K^{1−μ})(U_{i})}_{i∈I} be such that ϵ_{1}(d_{i}) = v_{i} (i ∈ I). Then b(K^{1−μ}, ρ) = (1 − μ){d_{i} − d_{j}}, which implies that

tb(K^{1−μ}, ρ) = (1 − μ){^{t}d_{i} − ^{t}d_{j}}.

On the other hand, {−^{t}d_{i} ∈ D_{S}^{≤1}(K^{μ})(U_{i})}_{i∈I} are local liftings of v, which means that b(K^{μ}, ρ) = −μ\{^{t}d_{i} − ^{t}d_{j}\} = \frac{μ}{μ−1} ^{t}b(K^{1−μ}, ρ). Thus

a(K^{μ}, ρ) = \frac{2μ − 1}{2μ}b(K^{μ}, ρ).  q.e.d.

Theorem 2.8. Replace M by an affine open set, and let {U_{i}}_{i∈I} be an affine open covering of S. Then, for any 0 ≠ v ∈ f^{∗}T_{M}(S), there are

d_{S}^{i} ∈ D_{S}^{≤1}(K^{μ})(U_{i}), d_{S/M}^{i} ∈ D_{S/M}^{≤1}(K^{μ})(U_{i}), D_{S/M}^{i} ∈ D_{S/M}^{≤2}(K^{μ})(U_{i}),

where K := K_{S/M}, such that

\{H(v)_{i} := d_{S}^{i} − d_{S/M}^{i} + \frac{2}{1 − 2μ}D_{S/M}^{i} ∈ D_{S}^{≤2}(K^{μ})(U_{i})\}_{i∈I}

form a global section H(v) ∈ H^{0}(S, D_{S}^{≤2}(K^{μ})) with

σ(H(v)) = v, ϵ_{2}(H(v)) = \frac{2}{1 − 2μ}δ_{H}(v).

Proof. Let {d_{S}^{i} ∈ D_{S}^{≤1}(K^{μ})(U_{i})}_{i∈I} be such that σ(d_{S}^{i}) = v|_{U_{i}}. Then

{μ(d_{S}^{i} − d_{S}^{j}) ∈ D_{S/M}^{≤1}(K^{μ})(U_{i} ∩ U_{j})}

defines the class b(K^{μ}, δ_{H}(v)) ∈ H^{1}(S, D_{S/M}^{≤1}(K^{μ})), which is the obstruc-

tion for lifting δ_{H}(v) ∈ H^{0}(S, S^{2}T_{S/M}) to H^{0}(S, S^{2}D_{S/M}^{≤1}(K^{μ})). Let

{D_{S/M}^{i} ∈ D_{S/M}^{≤2}(K^{μ})(U_{i})}_{i∈I}

be local liftings of δ_{H}(v). Then, by Proposition 2.7,

\{d_{S}^{i} − d_{S}^{j}\} = \frac{2}{2μ − 1}\{D_{S/M}^{i} − D_{S/M}^{j}\}

as cohomology classes. Thus there are {d_{S/M}^{i} ∈ D_{S/M}^{≤1}(K^{μ})(U_{i})}_{i∈I} satisfying the requirements in the theorem. q.e.d.

Corollary 2.9. There exists a unique projective heat operator,

H : T_{M} → f_{∗}W(Θ^{k})/O_{M}

(where M is global) such that (f_{∗}ϵ_{2}) · H : T_{M} → f_{∗}S^{2}T_{S/M} coincides with f_{∗}δ_{H}.

Proof. For any open set U ⊂ M and v ∈ T_{M}(U), by Theorem 2.8, we can construct a H(v) ∈ f_{∗}W(Θ^{k})(U). If H(v)′ is another such operator, H(v) − H(v)′ must have symbol in H^{0}(f^{−1}(U), T_{S/M}) = 0, so

H(v) − H(v)′ ∈ H^{0}(f^{−1}(U), O_{S}) = f_{∗}O_{S}(U) = O_{M}(U).

Hence a unique map T_{M} → f_{∗}W(Θ^{k})/O_{M}. q.e.d.

Now we construct the logarithmic extension of above operator. Let C̃ → M̃ be the family in the introduction. Let U(r, d) → M̃ be the family of moduli spaces of semistable (for canonical polarization) torsionfree sheaves of rank r and degree d. Fix a line bundle N on C̃ of relative degree d, let f : S → M be the family of moduli spaces of stable bundles of rank r with fixed determinant N |_{C} ([C] ∈ M).

Notation 2.10. Let f_{Z} : Z → M̃ be defined as the Zariski closure of S in U(r, d) and f_{T} : T → M̃ be the open set of Z consisting of locally free sheaves. Let f̃: S̃ ⊂ T → M̃ be the open set of stable bundles. Then f̃: S̃ → M̃ is smooth (cf. Lemma 4.4).

Let E be the unversal bundle on X̃, where X̃ is defined by

X̃ = C̃×M̃ S̃ ⟶^{π} S̃
↓  f̃↓
C̃  ⟶ M̃.

HITCHIN’S CONNECTION AND 2ND ORDER OPERATORS 317

Let D ⊂ X̃ be the divisor of singular curves. Then we have

G_{E} ⊂ ^{tr} A_{E}^{−1}(logD)

fitting into the exact sequence

0 → ω_{X̃/S̃} → G_{E} ⟶^{res} 𝔼 := End^{0}(E) → 0. (2.6)

Similarly, there is a sheaf S(G_{E}) ⊂ Sym^{2}(G_{E}) ⊗ ω_{X̃/S̃}^{−1} fitting into

0 → G_{E} ⟶^{ι} S(G_{E}) ⟶^{q} ω_{X̃/S̃}^{−1} → 0. (2.7)

They induce the following exact sequences:

0 → R^{1}π_{∗}(ω_{X̃/S̃}) → R^{1}π_{∗}(G_{E}) ⟶^{res} R^{1}π_{∗}(𝔼) → 0, (2.8)

0 → R^{1}π_{∗}(G_{E}) ⟶^{ι} R^{1}π_{∗}S(G_{E}) ⟶^{q} R^{1}π_{∗}(ω_{X̃/S̃}^{−1}) → 0. (2.9)

Let Δ̃ ⊂ X̃ ×S̃ X̃ := P be the diagonal and O(Δ̃) be the dual of its ideal sheaf. Then 0 → ω_{P/S̃} → ω_{P/S̃}(Δ̃) → Ext_{P}^{1} (OΔ̃, ω_{P/S̃}) → 0, one checks that the relative dualizing sheaf ω_{P/S̃} is ω_{X̃/S̃} ⊠ ω_{X̃/S̃} and Ext_{P}^{1} (OΔ̃, ω_{P/S̃}) is the relative dualizing sheaf of Δ̃/S̃. Thus we have 0 → O → O(Δ̃) → ω_{X̃/S̃}^{−1} → 0, (2.10) which similarly induces the commutative diagram

R^{1}π_{∗}(G_{E}) ⟶^{δ_{1}} R^{1}π_{∗}(G_{E})
ι↓  ^{Sym}^{2}↓
R^{1}π_{∗}S(G_{E}) \underrightarrow{δ̃} S^{2}R^{1}π_{∗}(G_{E})
q↓  ^{S^{2}(res )}↓
R^{1}π_{∗}(ω_{X̃/S̃}^{−1}) ⟶^{δ} S^{2}R^{1}π_{∗}(𝔼)

where δ_{1} denote the map induced by δ̃, the first vertical exact sequence is (2.9), the second vertical exact sequence is induced by taking the 2-nd symmetric tensor of (2.8) (note that OS̃ ≅ R^{1}π_{∗}(ω_{X̃/S̃})).

318 X. SUN & I-H. TSAI

Let B = M̃ \ M and W = f̃^{−1}(B) ⊂ S̃. Consider

0 → T_{S̃/M̃} → TS̃ ⟶^{df̃} f̃^{∗}TM̃ → 0,

0 → D_{S̃/M̃}^{≤1}(L) → D≤1 ; S̃(L) ⟶^{σ} f̃^{∗}TM̃ → 0.

Notation 2.11. Let TM̃(B) ⊂ T_{M} be the subsheaf of vector fields that preserve B. Let TS̃(log W) ⊂ TS̃, D≤1 ; S̃(L)(log W) ⊂ D≤1 ; S̃(L) be the subsheaves such that the following are exact sequences:

0 → T_{S̃/M̃} → TS̃(log W) ⟶^{df̃} f̃^{∗}TM̃(B) → 0, (2.11)

0 → D_{S̃/M̃}^{≤1}(L) → D≤1 ; S̃(L)(log W) ⟶^{σ} f̃^{∗}TM̃(B) → 0. (2.12)

Lemma 2.12. The map δ_{1} : R^{1}π_{∗}(G_{E}) → R^{1}π_{∗}(G_{E}) is identity. Proof. Since S ⊂ S̃ is dense in S̃ by definition (cf. Notation 2.10),

the lemma follows from Lemma 2.4. q.e.d. Theorem 2.13.

i) There is an isomorhism ϕ : R^{1}π_{∗}(G_{E}) ≅ D_{S̃/M̃}^{≤1}(λ_{𝔼}) such that the

following diagram is commutative:

0 ⟶ OS̃ =R^{1}π_{∗}ω_{X̃/S̃} ⟶ R^{1}π_{∗}(G_{E}) ⟶^{res} R^{1}π_{∗}(𝔼) ⟶ 0
(2r)·id↓ ^{ϕ}↓ _{ϑ}↓
0 ⟶ OS̃ =R^{0}π_{∗}ω_{X̃/S̃}[1] ⟶ D_{S̃/M̃}^{≤1}(λ_{𝔼}) ⟶^{ϵ_{1}} T_{S̃/M̃} ⟶ 0.

ii) For any affine covering {M̃_{i}}_{i∈I} of M̃, on each S̃_{i} := f̃^{−1}(M̃_{i}), there is an isomorphism ϕ_{i} : R^{1}π_{∗}S(G_{E}) ≅ D≤1 ; S̃(λ_{𝔼})(log W) such that the following diagram commutes on S̃_{i}

0 ⟶ R^{1}π_{∗}(G_{E}) ⟶^{2·ι} R^{1}π_{∗}S(G_{E}) ⟶^{q} R^{1}π_{∗}(ω_{X̃/S̃}^{−1}) ⟶ 0
ϕ↓ ^{ϕ_{i}}↓ ↓
0 ⟶ D_{S̃/M̃}^{≤1}(λ_{𝔼}) ⟶ D≤1 ; S̃(λ_{𝔼})(log W) ⟶^{σ} f̃^{∗}TM̃(B) ⟶ 0

and {ϕ_{i} − ϕ_{j}} defines a class in H^{1}(Ω1 ; M̃(log B)). Proof. See Proposition 4.5 and Theorem 4.8. q.e.d.

Similarly, we have the commutative diagram on each S̃_{i}

0 ⟶ D_{S̃/M̃}^{≤1}(λ_{𝔼}) ⟶ D≤1 ; S̃(λ_{𝔼})(log W) ⟶^{σ} f̃^{∗}TM̃(B) ⟶ 0
∥  δ̃_{H}^{i}↓  ^{δ_{H}}↓
0 ⟶ D_{S̃/M̃}^{≤1}(λ_{𝔼}) ⟶ S^{2}D_{S̃/M̃}^{≤1}(λ_{𝔼}) ⟶^{S^{2}ϵ_{1}} S^{2}T_{S̃/M̃} ⟶ 0.

Proposition 2.14. For any ρ = δ_{H}(v) ∈ H^{0}(S̃, S^{2}T_{S̃/M̃}), where M̃ is replaced by an affine open set and v ∈ H^{0}(S̃, f̃^{∗}TM̃(B)), one has:

i) 2a(L, ρ) = b(L, ρ) + ^{t}b(L^{−1} ⊗ K_{S̃/M̃}, ρ).

ii) When L = K_{S̃/M̃}^{μ}, where μ ∈ ℚ and μ ≠ 1, one has

a (K_{S̃/M̃}^{μ}, ρ) = \frac{2μ − 1}{2μ}b (K_{S̃/M̃}^{μ}, ρ) .

Proof. Note that we still have K = K_{S̃/M̃} = λ_{𝔼}, the proof is the same as that of Proposition 2.7 by using Theorem 2.13. We just remark two points:

(1) For any operator d of D≤1 ; S̃(L)(log W) ⊂ D≤1 ; S̃(L), its adjoint operator ^{t}d is still in D≤1 ; S̃(L^{−1} ⊗ K)(log W).

(2) For any nonzero μ ∈ ℚ, the canonical isomorphism ψ_{μ} : D≤1 ; S̃(K) ≅ D≤ ; S̃ (K^{μ}) induces an isomorphism D≤1 ; S̃ (K) (log W) ≅

D≤1 ; S̃(K^{μ})(log W). q.e.d.

Theorem 2.15. Replace M̃ by an affine open set, and let {U_{i}}_{i∈I} be an affine open covering of S̃. Then, for any v ∈ f̃^{∗}TM̃(B)(S̃), there are

di ; S̃ ∈ D≤1 ; S̃(K^{μ})(log W)(U_{i}), d_{S̃/M̃}^{i} ∈ D_{S̃/M̃}^{≤1}(K^{μ})(U_{i}),

and D_{S̃/M̃}^{i} ∈ D_{S̃/M̃}^{≤2}(K^{μ})(U_{i}) such that

\{H(v)_{i} := di ; S̃ − d_{S̃/M̃}^{i} + \frac{2}{1 − 2μ}D_{S̃/M̃}^{i} ∈ D≤2 ; S̃(K^{μ})(U_{i})\}_{i∈I}

form a global section H(v) ∈ H^{0}(S̃, D≤2 ; S̃(K^{μ})) with

σ(H(v)) = v, ϵ_{2}(H(v)) = \frac{2}{1 − 2μ}δ_{H}(v).

Proof. Let {di ; S̃ ∈ D≤1 ; S̃(K^{μ})(log W)(U_{i})}_{i∈I} be such that σ(di ; S̃) = v|_{U_{i}}. Then

{μ (di ; S̃ − dj ; S̃) ∈ D_{S̃/M̃}^{≤1}(K^{μ})(U_{i} ∩ U_{j})}

defines the class b(K^{μ}, δ_{H}(v)) ∈ H^{1}(S̃, D_{S̃/M̃}^{≤1}(K^{μ})), which is the obstruc-

tion for lifting δ_{H}(v) ∈ H^{0}(S̃, S^{2}T_{S̃/M̃}) to H^{0}(S̃, S^{2}D_{S̃/M̃}^{≤1}(K^{μ}). Let

{D_{S̃/M̃}^{i} ∈ D_{S̃/M̃}^{≤2}(K^{μ})(U_{i})}_{i∈I}

be local liftings of δ_{H}(v). Then, by Proposition 2.14,

\{di ; S̃ − dj ; S̃\} = \frac{2}{2μ − 1} \{D_{S̃/M̃}^{i} − D_{S̃/M̃}^{j}\}

as cohomology classes. Thus there are {d_{S̃/M̃}^{i} ∈ D_{S̃/M̃}^{≤1}(K^{μ})(U_{i})}_{i∈I} satisfying the requirements in the theorem. q.e.d.

Corollary 2.16. There exists a unique projective heat operator

H̃ : TM̃(B) → f̃_{∗}D≤2 ; S̃(Θ^{k})/OM̃

such that (f̃_{∗}ϵ_{2}) · H̃ : TM̃(B) → f̃_{∗}S^{2}T_{S̃/M̃} coincides with f̃_{∗}δ_{H}. Moreover, f̃_{∗}(Θ^{k}) is a coherent sheaf on M̃.

Proof. The coherence of f̃_{∗}(Θ^{k}) follows from Theorem 4.10. Since f̃_{∗}T_{S̃/M̃} = 0 is still true, the rest follows the same proof of Corollary 2.9 if f̃_{∗}OS̃ = OM̃. We will prove (cf. Proposition 4.9) that the fibres of f̃ : S̃ → M̃ is dense in the fibres of f_{Z} : Z → M̃. Thus Z \ S̃ has codimension at least 2. By passing to the normalization ι : Z̃ → Z, ι^{−1}(S̃) ≅ S̃ and codim(Z̃ \ι^{−1}(S̃)) ≥ 2 since S̃ is the open set of smooth points of Z and ι is finite. We have f̃_{∗}OS̃ = (f_{Z} · ι)_{∗}OZ̃ = OM̃. q.e.d.

3. First order differential operators of the determinant bundle

We give at first a short review of what we need from [2]. Let π : X → S be a smooth proper morphism of relative dimension 1 between smooth varieties in characteristic 0. We write K_{X/S} or ω_{X/S} interchangeably for the dualizing sheaf. One has an exact sequence

0 → T_{X/S} → T_{X} ⟶^{dπ} π^{∗}T_{S} → 0. (3.1)

As in [2], one defines the subsheaf π^{−1}T_{S} ⊂ π^{∗}T_{S} and its preimage T_{π} = dπ^{−1}T_{π} ⊂ T_{X}, defining the exact sequence

0 → T_{X/S} → T_{π} ⟶^{dπ} π^{−1}T_{S} → 0. (3.2)

Let E be a vector bundle on X, and λ_{E} = det Rπ_{∗}E be its determinant bundle. The Atiyah algebra A_{E} is the subalgebra of the sheaf of firstorder differential operators on E with symbolic part in (id ⊗ T_{X}) ≅ T_{X}. The relative Atiyah algebra A_{E/S} ⊂ A_{E} consists of those differential operators with symbolic part in T_{X/S}, and A_{E,π} ⊂ A_{E} with symbolic part in T_{π}. Let ^{tr} A_{E}^{−1} be the subquotient of the sheaf defined in [2]

E ⊠_{O_{S}} (E^{∗} ⊗ ω_{X/S})(2Δ)/E ⊠_{O_{S}} (E^{∗} ⊗ ω_{X/S})(−Δ)

where Δ ⊂ X ×_{S} X denotes the diagonal, which fits into an exact sequence

0 → ω_{X/S} → ^{tr} A_{E}^{−1} ⟶^{res} A_{E/S} → 0. (3.3)

The trace complex is defined by

trA_{E}^{•} : O_{X} ⟶^{d_{X/S}} ^{tr}A_{E}^{−1} ⟶^{res} A_{E,π} (3.4)

with A_{E,π} in degree 0. One has:

Proposition 3.1. ^{tr} A_{E}^{•} carries an algebra structure for which R^{0}π_{∗}(^{tr}A_{E}^{•}) is canonically isomorphic to A_{λ_{E}} ([2], 2.3.1, see also [4]).

For the purpose of this paper it is more convenient to define the trace complex concentrated only on i = −1 and i = 0 of the original trace complex. This modified trace complex is still denoted by ^{tr} A_{E}^{•} whose 0-th direct image is easily seen to be the same as that of the original one. With the modified trace complex, one has now an exact sequence

0 → ω_{X/S}[1] → ^{tr}A_{E}^{•} → A_{E,π}^{•} → 0, (3.5)

where the complex A_{E,π}^{•} is defined by

A_{E,π}^{•} : A_{E/S} → A_{E,π}, (3.6)

and thus is quasi-isomorphic to π^{−1}T_{S}.

Notation 3.2. Let π : X → S be as before, and f : S → M be a smooth morphism where M is a smooth variety. Denote by

A_{E,π/M} ⊂ A_{E,π}, T_{π/M} ⊂ T_{X/M} ⊂ T_{π}

the pullback of π^{−1}T_{S/M} ⊂ π^{−1}T_{S}. Let

trA_{E/M}^{•} := (^{tr} A_{E}^{−1} → A_{E,π/M}), A_{E,π/M}^{•} := (A_{E/S} → A_{E,π/M}).

Proposition 3.3 ([2], Theorem 2.3.1). The exact sequences

0 → ω_{X/S}[1] → ^{tr}A_{E}^{•} → A_{E,π}^{•} → 0
0 → ω_{X/S}[1] → ^{tr} A_{E/M}^{•} → A_{E,π/M}^{•} → 0

have 0-th direct images (via π) isomorphic to

0 → O_{S} → D_{S}^{≤1}(λ_{E}) → T_{S} → 0
0 → O_{S} → D_{S/M}^{≤1}(λ_{E}) → T_{S/M} → 0.

Furthermore, we need to review the description of ^{tr} A_{E}^{•} in terms of local coordinates, [2], p. 660. Let t be a local coordinate (along the fiber), and a trivilization I : O_{X}^{n} ≅ E; s a local coordinate on S. Note t naturally induces local coordinates (t_{1}, t_{2}) around the diagonal of X ×_{S} X. One has isomorphisms via t and I,

(3.7)  (t, I)_{−1} : O_{X} ⊕ Mat_{n}(O_{X}) ⊕ O_{X}

≅ ^{tr} A_{E}^{−1}, (t, I)_{−1}(χ, B, ν) = (χ, B, ν)_{(t,I)}

= I [\frac{χ(t_{1})}{(t_{2} − t_{1})^{2}} + \frac{B(t_{1})}{(t_{2} − t_{1})} + ν(t_{1})] dt_{2};

(t, I)_{0} : T_{π} ⊕ Mat_{n}(O_{X})

≅ ^{tr} A_{E}^{0} = A_{E,π}, (t, I)_{0}(τ, A)

= (τ, A)_{I} = (t, I)_{0}[τ(t, s)∂_{t} + μ(s)∂_{s} + A].

For different choices of coordinates and trivializations, one has the gauge change and coordinate change formulas: with g ∈ GL_{r}(O_{X}) and y = y(t) (the notation below being slightly different from that of [2])

(3.8) (τ, A) := α ⟶^{g} (τ, −τ(g)g^{−1} + gAg^{−1}) := β(⇔ α_{Ig} = β_{I});
(χ, B, ν) ⟶^{g} (χ, −χg′g^{−1} + gBg^{−1},

Tr( − \frac{1}{2}χg′′g^{−1} + χ(g′g^{−1})^{2} − Bg^{−1}g′) + ν);

(χ, B, ν) ⟶^{y(t)} (χy′^{−1}, B, rχ(\frac{1}{6} \frac{y′′′}{y′^{2}} − \frac{1}{4} \begin{matrix}y′′ & 2 \\ y′ & 3 \end{matrix} ) + \frac{1}{2} \frac{y′′}{y′} TrB + νy′).

The main result of this section is Theorem 3.7 which enables us to take care of Theorem 2.5. Following the notation of Section 2, let

p : X = C ×_{M} S → C

be the projection.

Definition 3.4. End(E)^{−1} := res^{−1}(End(E)) ⊂ ^{tr}A_{E}^{−1}, and

G_{E} := res^{−1}(𝔼) ⊂ End(E)^{−1}

where End(E) = 𝔼 ⊕ O_{X} with its trace free part 𝔼 := End^{0}(E) and the trivial bundle O_{X}. There are exact sequences

0 → ω_{X/S} → G_{E} ⟶^{res} 𝔼 → 0;

0 → ω_{X/S} → End(E)^{−1} ⟶^{res} End(E) → 0.

Consider the natural morphism

S^{2}(G_{E}) ⊗_{O_{X}} T_{X/S} ⟶^{q} S^{2}(𝔼) ⊗_{O_{X}} T_{X/S} → 0

induced by G_{E} ⟶^{res} 𝔼. Denote the kernel of q by K. There is a canonical isomorphism ι : G_{E} ≅ K, that is ι(s) = Sym^{2}(dt ⊗ s) ⊗ ∂_{t} locally.

Definition 3.5. q^{−1}(id ⊗ T_{X/S}) := S(G_{E}) where id ∈ S^{2}(𝔼) is the identity element. It follows that we have the exact sequence

0 → G_{E} ⟶^{ι} S(G_{E}) → T_{X/S} → 0. (3.9)

Locally, for choosen coordinate and trivilization (cf. (3.7)), any local section s ∈ S(G_{E}) is of the form

s = ( \begin{matrix}χ ∑_{a}(0, J_{a}, 0) ⊗ (0, J_{a}, 0) + ∑_{a} μ_{a}(0, J_{a}, 0) ⊗ (0, 0, 1) \\ + ∑_{a} ν_{a}(0, 0, 1) ⊗ (0, J_{a}, 0) + w(0, 0, 1) ⊗ (0, 0, 1) \end{matrix}) ⊗ ∂_{t}

where J_{a} is a (local) basis of 𝔼 (which we assume to be orthonormal under Trace( ·)) such that ∑_{a} J_{a} ⊗ J_{a} = id (such a canonical choice of J_{a} will simplify computations later).

We will need the Kodaira–Spencer maps

KS_{S} : T_{S} → R^{1}π_{∗}A_{E/S}^{0}, A_{E/S}^{0} = A_{E/S}/O_{X}; (3.10)

KS_{M} : f^{∗}T_{M} → R^{1}π_{∗}T_{X/S}.

They fit into the following commutative diagram:

0 ⟶ R^{1}π_{∗}(𝔼) ⟶ R^{1}π_{∗}A_{E/S}^{0} ⟶ R^{1}π_{∗}T_{X/S} ⟶ 0
KS↑  ^{KS_{S}}↑  ^{KS_{M}}↑
0 ⟶ T_{S/M} ⟶  T_{S}  ⟶ f^{∗}T_{M} ⟶ 0.

Remark 3.6. One way to see KS_{S} of (3.10) is via the natural map (with cohomology) A_{E,π}^{•} → A_{E/S}^{0}[1] from (3.6); similarly KS_{M} is via (T_{C/M} → T_{π}) → T_{C/M}[1] (with cohomology) combined with its pullback via f : S → M. The diagram (3.10) via natural maps T_{S} → f^{∗}T_{M} and R^{1}π_{∗}A_{E/S}^{0} → R^{1}π_{∗}T_{X/S}, for Kodaira–Spencer maps commute.

Theorem 3.7.

i) If the Kodaira–Spencer map KS : T_{S/M} → R^{1}π_{∗}(𝔼) is an isomorphism, then there exists a canonical isomorphism

ϕ : R^{1}π_{∗}(G_{E}) ≅ R^{0}π_{∗}(End(𝔼)^{−1} → A_{𝔼,π/M}) ≅ R^{0}π_{∗} ^{tr} A_{𝔼/M}^{•}.

ii) If KS_{S} is an isomorphism and shrink M enough, then the above ϕ extends to an isomorphism

ϕ = ϕ_{−λ} : R^{1}π_{∗}S(G_{E}) ≅ R^{0}π_{∗}(^{tr} A_{𝔼}^{−1} → A_{𝔼,π}).

(The right-hand sides of i), ii) are canonically identified with D_{S/M}^{≤1}(λ_{𝔼}),

D_{S}^{≤1}(λ_{𝔼}) respectively, cf. Proposition 3.3.)

Remark 3.8.

R^{0}π_{∗}(End(𝔼)^{−1} → A_{𝔼,π/M}) ≅ R^{0}π_{∗}(End(𝔼)^{−1} → A_{𝔼,π})

holds under A_{𝔼,π/M} → A_{𝔼,π}, cf. the proof of i) of Proposition 4.5.

Corollary 3.9. Assumptions being as in ii) of Theorem 3.7, suppose λ_{𝔼} ≅ Θ^{−λ} (λ = 2(r, d)). For k ∈ ℤ one has an isomorphism denoted by

ϕ_{k} : R^{1}π_{∗}S(G_{E}) ≅ D_{S}^{≤1}(Θ^{k}),

extending Theorem 3.7. (If k = −λ, write ϕ for ϕ_{−λ}.)

Proposition 3.10.

i) The morphism 𝔼 → End^{0}(𝔼) induced by the adjoint representation extends naturally to a canonical morphism ad : A_{E} → A_{𝔼} (preserving algebra structures).

ii) The morphism ad has a natural lifting ãd : G_{E} → G_{𝔼}, which induces (2r) · id on ω_{X/S}.

Proof.

i) For any D ∈ A_{E}, L ∈ 𝔼, ad(D)(L) := D ∘ L − L ∘ D is a section of 𝔼 (note that Tr(D ∘ L − L ∘ D) = ϵ_{1}(D)Tr(L)). Thus ad(D) defines a map 𝔼 → 𝔼, which is a differential operator since ad(D)(λ · L) = λ · ad(D)(L) + ϵ_{1}(D)(λ) · L.

ii) This can be proved via the local formulas given in (3.7) Namely, a local element of G_{E} is expressed as (0, B, ν) (with B ∈ Mat_{r}(O_{X}) and ν ∈ O_{X}). Define a lifting by sending (0, B, ν) to (0, adB, 2rν). We will show by using formulas in (3.8) that the above lifting is in fact globally defined. Using χ = 0, TrB = 0 and Tr(adB) = 0 in (3.8), we have

(3.11) (0, B, ν) ⟶^{g} (0, gBg^{−1}, Tr(−Bg^{−1}g′) + ν);
(0, B, ν) ⟶^{y(t)} (0, B, νy′);
(0, adB, ν) ⟶^{g} (0, ad(gBg^{−1}), Tr( − adB ad(g^{−1}g′)) + ν);
(0, adB, ν) ⟶^{y(t)} (0, adB, νy′).

If change the trivilization of E by g, the induced trivilization of End(E) will be changed by e_{g} : 𝕄_{r}(O_{X}) → 𝕄_{r}(O_{X}), where e_{g}(B) = gBg^{−1}. It is easy to check that e_{g}^{−1}e′_{g} = ad(g^{−1}g′), thus we obtain the term Tr( − adB ad(g^{−1}g′)) in the 3rd row above. One knows that

(3.12) Tr(adM adN) = 2rTr(MN)

for traceless matrices M, N of rank r. Let (g^{−1}g′)_{0} be the traceless compoment of g^{−1}g′. Note TrB = 0, ad(g^{−1}g′) = ad((g^{−1}g′)_{0}), and Tr(Bg^{−1}g′) = Tr(B(g^{−1}g′)_{0}) etc. It follows that the morphism (0, B, ν) → (0, adB, 2rν) as given is well-defined (globally). q.e.d.

Lemma 3.11. R^{0}π_{∗}A_{E,π} ≅ O_{S} and R^{0}π_{∗}A_{E,π}^{0} = 0 provided KS_{S} being injective, where A_{E,π}^{0} := A_{E,π}/O_{X}.

Proof. It suffices to prove, by using an exact sequence similar to (3.2) (with T^{•} replaced by A^{•}), that:

i) π_{∗}π^{−1}T_{S} → R^{1}π_{∗}A_{E/S} is injective.

ii) π_{∗}A_{E/S} ≅ O_{S}.

But the map in i) composed with R^{1}π_{∗}A_{E/S} → R^{1}π_{∗}A_{E/S}^{0} is nothing but KS_{S}, hence i). ii) is from π_{∗}T_{X/S} = 0 (genus ≥ 2) and π_{∗}End(E) = O_{S} (E is fiberwise stable). q.e.d.

326 X. SUN & I-H. TSAI

Proof of Theorem 3.7 i). First, we remark that morphisms in Proposition 3.10 make the following diagram of complexes commutative:

(^{tr} A_{𝔼}^{−1} ⟶^{res} A_{𝔼,π/M}) ⟶ (A_{𝔼/S} → A_{𝔼,π/M})
↑  _{ad}↑
(G_{E} ⟶^{res} A_{E,π/M}) ⟶ (A_{E/S}^{0} → A_{E,π/M}^{0}).

Second, we observe that the commutative diagram

0 ⟶ 𝔼 ⟶ A_{E,π/M}^{0} ⟶ T_{π/M} ⟶ 0
∥ ↑ ↑
0 ⟶ 𝔼 ⟶ A_{E/S}^{0} ⟶ T_{X/S} ⟶ 0

induces a commutative diagram

T_{S/M}  π_{∗}T_{π/M} ⟶^{KS} R^{1}π_{∗}(𝔼) ⟶ R^{1}π_{∗}A_{E,π/M}^{0}
↑ ∥ ↑
0 ⟶ R^{1}π_{∗}(𝔼) ⟶ R^{1}π_{∗}A_{E/S}^{0}.

Thus R^{1}π_{∗}(𝔼) vanishes in R^{1}π_{∗}A_{E,π/M}^{0} since KS is an isomorphism.

We construct ϕ for any affine open set U^{i} ⊂ S. Let {U_{i}, Ẋ_{i}} be an affine covering of π^{−1}(U^{i}), let U̇_{i} = U_{i} ∩ Ẋ_{i}. For any Čech cocycle rU̇_{i} ∈ G_{E}(U̇_{i}) in C^{1}(G_{E}), the class [res(rU̇_{i})] ∈ R^{1}π_{∗}(𝔼)(U^{i}) vanishes in R^{1}π_{∗}A_{E,π/M}^{0}(U^{i}). Thus there exists τẊ_{i} ∈ A_{E,π/M}^{0}(Ẋ_{i}), τ_{U_{i}} ∈ A_{E,π/M}^{0}(U_{i}) such that τẊ_{i} − τ_{U_{i}} = res rU̇_{i} on U̇_{i}. For given rU̇_{i}, the choice of τẊ_{i} and τ_{U_{i}} is unique (by Lemma 3.11). Then {ad(τẊ_{i}), ad(τ_{U_{i}}); ãd(rU̇_{i})} (3.13) is a Čech cocycle in C^{0}(G_{𝔼} → A_{𝔼,π/M}) (cf. [2], p. 673). It is easily checked that the assignment (3.14) ϕ̃: rU̇_{i} → {ad(τẊ_{i}), ad(τ_{U_{i}}); ãd(rU̇_{i})} ∈ C^{0}(^{tr} A_{𝔼/M}^{•}) preserves the respective coboundaries. Hence it descends to a map

ϕ : R^{1}π_{∗}(G_{E}) → R^{0}π_{∗} ^{tr}A_{𝔼/M}^{•}.

In the same way, we construct ϑ : R^{1}π_{∗}(𝔼) → R^{0}π_{∗}A_{𝔼,π/M}^{•} such that

0 ⟶ R^{1}π_{∗}ω_{X/S} ⟶ R^{1}π_{∗}(G_{E}) ⟶^{res} R^{1}π_{∗}(𝔼) ⟶ 0
(2r)·id↓  ^{ϕ}↓  _{ϑ}↓
0 ⟶ R^{0}π_{∗}ω_{X/S}[1] ⟶ R^{0}π_{∗} ^{tr} A_{𝔼/M}^{•} ⟶^{res} R^{0}π_{∗}A_{𝔼,π/M}^{•} ⟶ 0

is commutative. The map ϑ is the composition of

KS^{−1} : R^{1}π_{∗}(𝔼) → T_{S/M} = R^{0}π_{∗}(A_{E/S}^{0} → A_{E,π/M}^{0})

and the map R^{0}π_{∗}(A_{E/S}^{0} → A_{E,π/M}^{0}) → R^{0}π_{∗}A_{𝔼,π/M}^{•}, which is induced by the qusi-isomorphism ad of complexes at the begining of our proof (it is a qusi-isomorphism since ad(D) has the same symbol with D ∈ A_{E}). Thus ϑ is an isomorphism, then ϕ has to be an isomorphism. q.e.d.

Both R^{1}π_{∗}(G_{E}) and R^{0}π_{∗}(End(𝔼)^{−1} → A_{𝔼,π/M}) = R^{0}π_{∗} ^{tr} A_{𝔼/M}^{•} define extension classes in H^{1}(S, Ω_{S/M}). One has (by the preceding proof combined with proof of Proposition 3.10 for the constant 2r):

Corollary 3.12. The extension classes (e.c.[•] for short) satisfy

e.c.[R^{1}π_{∗}(G_{E})] = \frac{1}{2r} e.c.[D_{S/M}^{≤1}(λ_{𝔼})]

in H^{1}(S, Ω_{S/M}), where D_{S/M}^{≤1}(λ_{𝔼}) = R^{0}π_{∗}(End(𝔼)^{−1} → A_{𝔼,π/M}).

For ii) of the theorem, our proof will need the following result:

Proposition 3.13. There exisits r̃es : S(G_{E}) → A_{E/S}^{0} such that

0 ⟶ ω_{X/S} ⟶ S(G_{E}) \underrightarrow{r̃es} A_{E/S}^{0} ⟶ 0
↓  ∥  ↓
0 ⟶ G_{E} ⟶^{ι} S(G_{E}) ⟶ T_{X/S} ⟶ 0

is commutative. If KS_{S} in (3.10) is an isomorphism, then

0 → O_{S} → R^{1}π_{∗}S(G_{E}) → T_{S} → 0.

Proof. For any local section s ∈ S(G_{E}) in Definition 3.5, we define

r̃es(s) = (χ∂_{t}, \frac{1}{2} ∑_{a}(μ_{a} + ν_{a})J_{a}),

which is independent of the choice of {J_{a}}, thus well-defined locally. To show it well-defined globally, we need to check the invariance under gauge and coordinate changes. The invariance under local coordinate changes is straightforward. Under the gauge change g ∈ GL_{r}(O_{X}), the section s ∈ S(G_{E}) in Definition 3.5 becomes s_{g} ⊗ ∂_{t}, where s_{g} is

χ ∑_{a}(0, gJ_{a}g^{−1}, 0)^{⊗2} + ∑_{a}(μ_{a} − χTr(J_{a}g^{−1}g′))(0, gJ_{a}g^{−1}, 0) ⊗ (0, 0, 1)

+ ∑_{a}(ν_{a} − χTr(J_{a}g^{−1}g′))(0, 0, 1) ⊗ (0, gJ_{a}g^{−1}, 0)

+ (χ ∑_{a} Tr(J_{a}g^{−1}g′)^{2} − ∑_{a}(μ_{a} + ν_{a})Tr(J_{a}g^{−1}g′) + w) (0, 0, 1)^{⊗2}.

Then r̃es (s_{g} ⊗ ∂_{t}) = (χ∂_{t}, \frac{1}{2} ∑_{a} (μ_{a} + ν_{a} − 2χ Tr (J_{a}g^{−1}g′)) gJ_{a} g^{−1}) coincides with (χ∂_{t}, −χ∂_{t}(g)g^{−1} + \frac{1}{2} ∑_{a}(μ_{a} + ν_{a})gJ_{a}g^{−1}) in A_{E/S}^{0} = A_{E/S}/O_{X} since χ∂_{t}(g)g^{−1} = χg′g^{−1} = ∑_{a} χTr(J_{a}g^{−1}g′))gJ_{a}g^{−1} modulo O_{X}. Thus r̃es is gauge invariant and defined globally. Then the rest of this proposition is obvious. q.e.d.

Remark 3.14.

i) Sym^{2}(a ⊗ b) = \frac{1}{2}(a ⊗ b + b ⊗ a) for a, b ∈ G_{E}.

ii) Using Proposition 3.13 and assuming R^{1}f_{∗}O_{S} = 0 one has a quick interpretation of ii) of Theorem 3.7: note both R^{1}π_{∗}S(G_{E}) and R^{0}π_{∗} ^{tr} A_{𝔼}^{•} ≅ D_{S}^{≤1}(λ_{𝔼}) contain a subsheaf R^{1}π_{∗}(G_{E}) ≅ D_{S/M}^{≤1}(λ_{𝔼}). Given two extensions F, F′ of T_{S} by O_{S} suppose their subsheafs with symbolic part in T_{S/M} are isomorphic. Then F ≅ F′ (noncanonically) provided R^{1}f_{∗}O_{S} = 0 (since

H^{1}(S, Ω_{S}) → H^{1}(S, Ω_{S/M})

is injective if R^{1}f_{∗}O_{S} = 0 (and M is affine)). Theorem 3.7 ii) just proves an isomorphism of this kind without reference to R^{1}f_{∗}O_{S} = 0.

In what follows we will choose an open covering of C (shrink M if necessary), fix a local coordinate (along the fiber) on each open set of the covering and use the local description (3.7). Recall that

X = C ×_{M} S ⟶^{π} S
p↓  ^{f}↓
C  ⟶ M.

To simplify notation, we will cover C by two affine open sets V and Ċ. Let V ⊂ C be an affine open set and t be a local coordinate (along the fiber) on V. Let Ċ ⊂ C be an affine open set containing C \ V. Then we choose and fix the local coordinates (along the fibre) on V and Ċ. Let U = p^{−1}(V), Ẋ = p^{−1}(Ċ) (they are not affine but affine over S). We start with the construction of γ_{𝔼,U} : S(G_{E})|_{U} → ^{tr} A_{𝔼}^{−1}|_{U} (resp. γ_{𝔼,Ẋ} : S(G_{E})|Ẋ → ^{tr} A_{𝔼}^{−1}|Ẋ) such that the following diagram is commutative over U (resp. over Ẋ):

0 ⟶ ω_{X/S} ⟶ ^{tr} A_{𝔼}^{−1} ⟶^{res} A_{𝔼/S} ⟶ 0
r·id↑  ^{γ_{𝔼,U}} ↑  _{ad}↑
0 ⟶ ω_{X/S} ⟶ S(G_{E}) \underrightarrow{r̃es} A_{E/S}^{0} ⟶ 0
↓  ∥  ↓
0 ⟶ G_{E} ⟶^{ι} S(G_{E}) ⟶ T_{X/S} ⟶ 0

(∗)

where ad : A_{E/S}^{0} → A_{𝔼/S} is induced by the morphism in Proposition 3.10 i) that maps O_{X} to zero. Note that, except γ_{𝔼,U} (resp. γ_{𝔼,Ẋ}), other morphisms in the diagram are well-defined over the global M (i.e., need not shrink M).

Let {U^{i}}_{i} be an affine covering of S. Then {U_{i} := U ∩π^{−1}(U^{i})}_{i} (resp. {Ẋ_{i} = Ẋ∩π^{−1}(U^{i})}_{i}) is an affine covering of U (resp. Ẋ). We will define γ_{𝔼,U} (resp. γ_{𝔼,Ẋ}) on each U_{i} (resp. Ẋ_{i}) using the local description (3.7), then prove it is defined globally on U (resp. Ẋ). It is important to remark that on each U_{i} (resp. Ẋ_{i}) we use the local coordinate pulling back from V (resp. Ċ). Thus, to prove it is well-defined on U (resp. Ẋ), we only need to check the independence of trivialization of E over U_{i} (resp. Ẋ_{i}). On each U_{i}, fix a trivilization I of E on U_{i} and use the pullback coordinate (say, t) of V, we define the morphism by using the local description (3.7). For any local section α ∈ S(G_{E})(U_{i}), with respect to I and t

α = ( \begin{matrix}χ ∑_{a}(0, J_{a}, 0) ⊗ (0, J_{a}, 0) + ∑_{a} μ_{a}(0, J_{a}, 0) ⊗ (0, 0, 1) \\ + ∑_{a} ν_{a}(0, 0, 1) ⊗ (0, J_{a}, 0) + w(0, 0, 1) ⊗ (0, 0, 1) \end{matrix}) ⊗ ∂_{t}

(see the expression in Definition 3.5), one defines, where r = rank (E),

γ_{𝔼,U}(α) = (χ, \frac{1}{2} ∑_{a}(μ_{a} + ν_{a})adJ_{a}, rw)_{(t,I)} ∈ ^{tr} A_{𝔼}^{−1}(U_{i}), (3.15)

where one uses the natural trivilization I of 𝔼 induced from that of E.

Lemma 3.15. The assignment α → γ_{𝔼,U}(α) constructed above is bundle gauge-invariant; More precisely:

i) Under another choice of trivialization of E (on U_{i}), such that J_{a} → gJ_{a}g^{−1} so that α → α_{g}, the assignment to α_{g} becomes equal to the e_{g}-transformation of γ_{𝔼,U}(α), where e_{g} is the gauge of 𝔼 induced by g (cf. Proposition 3.10).

ii) Under a coordinate change y = y(t), the difference of γ_{𝔼,U}(α) is

(0, 0, rχ(\frac{1}{6} \frac{y′′′}{y′^{2}} − \frac{1}{4} \begin{matrix}y′′ & 2 \\ y′ & 3 \end{matrix} ))_{(t,I)} ∈ ω_{X/S}(U).

If β ∈ G_{E}(U_{i}), then γ_{𝔼,U}(ι(β)) = \frac{1}{2} ãd(β), where ãd : G_{E} → G_{𝔼} is the morphism in ii) of Proposition 3.10.

Proof. The proof is to check a straightforward commutativity relation caused by the gauge change, hence we simplify the notation if there is no danger of confusion. As in the proof of Proposition 3.13, using the explicit expression of α_{g} = s_{g} ⊗ ∂_{t}, one has

γ_{𝔼,U}(α_{g}) = (χ, \frac{1}{2} ∑_{a}(μ_{a} + ν_{a} − 2χTr(J_{a}g^{−1}g′))ad(gJ_{a}g^{−1}),

rχ ∑_{a} Tr(J_{a}g^{−1}g′)^{2} − r ∑_{a}(μ_{a} + ν_{a})Tr(J_{a}g^{−1}g′) + rw).

The goal is to show γ_{𝔼,U}(α_{g}) equals the e_{g}-transformation γ_{𝔼,U}(α)^{g} of γ_{𝔼,U}(α), which is

( \begin{matrix}& χ, −χe′_{g}e_{g}^{−1} + \frac{1}{2} ∑_{a}(μ_{a} + ν_{a})e_{g}ad(J_{a})e_{g}^{−1}, rw + \\ Tr & (−\frac{1}{2}χe′′_{g}e_{g}^{−1} + χ(e′_{g}e_{g}^{−1})^{2} − \frac{1}{2} ∑_{a}(μ_{a} + ν_{a})ad(J_{a})e_{g}^{−1}e′_{g}) \end{matrix})

Recall that e_{g} : 𝕄_{r}(O_{X}) → 𝕄_{r}(O_{X}) means e_{g}(B) = gBg^{−1}, we have e_{g}ad(J_{a})e_{g}^{−1} = ad(gJ_{a}g^{−1}). Thus the second components of γ_{𝔼,U}(α_{g}) and γ_{𝔼,U}(α)^{g} will coincide if e′_{g}e_{g}^{−1} = ∑_{a} Tr(J_{a}g^{−1}g′)ad(gJ_{a}g^{−1}), which is true since e′_{g}e_{g}^{−1} = ad(g′g^{−1}), e_{g}^{−1}e′_{g} = ad(g^{−1}g′). To finish the proof, we will show that their third components coincide. Since

\frac{1}{2} ∑_{a}(μ_{a} + ν_{a})Tr(ad(J_{a})e_{g}^{−1}e′_{g}) = r ∑_{a}(μ_{a} + ν_{a})Tr(J_{a}g^{−1}g′),

it will be done if one can show the following identity:

r ∑_{a} Tr(J_{a}g^{−1}g′)^{2} = Tr((e′_{g}e_{g}^{−1})^{2} − \frac{1}{2}e′′_{g}e_{g}^{−1}). (3.16)

Write e′′_{g}e_{g}^{−1} = (e′_{g}e_{g}^{−1})′ − e′_{g}(e_{g}^{−1})′ = (e′_{g}e_{g}^{−1})′ + e′_{g}e_{g}^{−1}e′_{g}e_{g}^{−1}, then

Tr(e′′_{g}e_{g}^{−1}) = Tr((e′_{g}e_{g}^{−1})^{2}) = Tr(ad(g′g^{−1})ad(g′g^{−1}))

(using (e_{g}e_{g}^{−1})′ = 0 and Tr((e′_{g}e_{g}^{−1})′) = Tr(e′_{g}e_{g}^{−1})′ = 0 here). Let (g′g^{−1})_{0} be the traceless part of g′g^{−1}. Then R.H.S. of (3.16) equals

\frac{1}{2}Tr(ad(g′g^{−1})ad(g′g^{−1})) = rTr((g′g^{−1})_{0}(g′g^{−1})_{0}).

By the choice of {J_{a}}_{a}, we have (g^{−1}g′)_{0} = ∑_{a} Tr(J_{a}g^{−1}g′)J_{a}. Then L.H.S of (3.16) equals rTr((g^{−1}g′)_{0}g^{−1}g′) = rTr((g^{−1}g′)_{0}(g^{−1}g′)_{0}). Thus (3.16) is true since g(g^{−1}g′)_{0}g^{−1} = (g′g^{−1})_{0}. We are done with i). The assertion ii) is a straightforward calculation using the coordinate change formula (3.8) and ∂_{y} = y′(t)^{−1}∂_{t}. For β = (0, ∑_{a} ξ_{a}J_{a}, w) ∈ G_{E}(U_{i}), ι(β) = \frac{1}{2}((0, ∑_{a} ξ_{a}J_{a}, w) ⊗ (0, 0, 1) + (0, 0, 1) ⊗ (0, ∑_{a} ξ_{a}J_{a}, w)) ⊗ ∂_{t}

= ( \begin{matrix}∑_{a} \frac{1}{2}ξ_{a}(0, J_{a}, 0) ⊗ (0, 0, 1) + ∑_{a} \frac{1}{2}ξ_{a}(0, 0, 1) ⊗ (0, J_{a}, 0) \\ + w(0, 0, 1) ⊗ (0, 0, 1) \end{matrix}) ⊗ ∂_{t}.

Thus γ_{𝔼,U}(ι(β)) = \frac{1}{2}(0, ad(∑_{a} ξ_{a}J_{a}), 2rw) = \frac{1}{2} ãd(β). q.e.d.

We have constructed γ_{𝔼,U} (the construction of γ_{𝔼,Ẋ} is similar) such that the above diagram (∗) is commutative over U (resp. over Ẋ). However, on U̇ := U ∩ Ẋ, the difference (caused by the difference of coordinates on U and Ẋ), γU̇ := γ_{𝔼,U} − γ_{𝔼,Ẋ} vanishes on G_{E}|U̇, but may not vanish on S(G_{E})|U̇ (cf. ii) of Lemma 3.15). It defines a morphism γU̇ : S(G_{E})|U̇ → ω_{X/S}|U̇ that induces γ̃U̇ : T_{X/S}|U̇ → ω_{X/S}|U̇, i.e., a section γ̃U̇ ∈ Hom(T_{X/S}, ω_{X/S})(U̇) = ω_{X/S}^{2}(p^{−1}(V̇)), where V̇ := V ∩ Ċ. The [γ̃U̇] defines a class of H^{1}(C, p_{∗}ω_{X/S}^{2}) = H^{1}(C, ω_{C/M}^{2}), which vanishes when M is replaced by an affine open set. Thus there exist

ψ̃_{U} ∈ Hom(T_{X/S}, ω_{X/S})(U), ψ̃Ẋ ∈ Hom(T_{X/S}, ω_{X/S})(Ẋ)

such that γ̃U̇ = ψ̃_{U} − ψ̃Ẋ. Let ψ_{U}, ψẊ denote the induced morphisms

ψ_{U} : S(G_{E})|_{U} → S(G_{E})/G_{E}|_{U} ≅ T_{X/S}|_{U} \underrightarrow{ψ̃_{U}} ω_{X/S}|_{U}
ψẊ : S(G_{E})|Ẋ → S(G_{E})/G_{E}|Ẋ ≅ T_{X/S}|Ẋ \underrightarrow{ψ̃Ẋ} ω_{X/S}|Ẋ.

Then it is easy to see that γU̇ = γ_{𝔼,U} − γ_{𝔼,Ẋ} = ψ_{U} − ψẊ. Let

β_{U} := γ_{𝔼,U} − ψ_{U}, βẊ := γ_{𝔼,Ẋ} − ψẊ.

Thus, by shrinking M, we have proved:

Proposition 3.16. The β_{U} and βẊ define a morphism

β : S(G_{E}) → ^{tr} A_{𝔼}^{−1},

which induces (through ι) \frac{1}{2} ãd on G_{E}, such that the following diagram is commutative:

0 ⟶ ω_{X/S} ⟶ ^{tr} A_{𝔼}^{−1} ⟶^{res} A_{𝔼/S} ⟶ 0
r·id↑  ^{β}↑  _{ad}↑
0 ⟶ ω_{X/S} ⟶ S(G_{E}) \underrightarrow{r̃es} A_{E/S}^{0} ⟶ 0
↓  ∥  ↓
0 ⟶ G_{E} ⟶^{ι} S(G_{E}) ⟶ T_{X/S} ⟶ 0.

Proof of Theorem 3.7 ii). The proof is similar to that of i) of the theorem. By

0 → A_{E/S}^{0} → A_{E,π}^{0} → π^{−1}T_{S} → 0 (3.17)

and by identifying its connecting map T_{S} → R^{1}π_{∗}A_{E/S}^{0} with the Kodaira–Spencer map KS_{S} (which will be treated more generally for log D in Proposition 4.5), we see that R^{1}π_{∗}A_{E/S}^{0} vanishes in R^{1}π_{∗}A_{E,π}^{0} if KS_{S} is an isomorphism. Thus, for any α ∈ S(G_{E})(U̇_{i}), there exist τẊ_{i} ∈ A_{E,π}^{0}(Ẋ_{i}) and τ_{U_{i}} ∈ A_{E,π}^{0}(U_{i}) such that τẊ_{i} − τ_{U_{i}} = r̃es(α). Then, by Proposition 3.16, we see that res(β(α)) = ad(r̃es(α)). Thus

ϕ̃(α) := {ad(τẊ_{i}), ad(τ_{U_{i}}); β(α)}

is a cocycle in C^{0}(^{tr} A_{𝔼}^{−1} → A_{𝔼,π}). It is clear that ϕ̃ induces a morphism

ϕ : R^{1}π_{∗}S(G_{E}) → R^{0}π_{∗}(^{tr} A_{𝔼}^{−1} → A_{𝔼,π}) = R^{0}π_{∗} ^{tr} A_{𝔼}^{•}.

Similarly, we construct ϑ : R^{1}π_{∗}A_{E/S}^{0} → R^{0}π_{∗}A_{𝔼,π}^{•}, which is the composition of

KS_{S}^{−1} : R^{1}π_{∗}A_{E/S}^{0} → T_{S} = R^{0}π_{∗}(A_{E/S}^{0} → A_{E,π}^{0})

and the isomorphism R^{0}π_{∗}(A_{E/S}^{0} → A_{E,π}^{0}) → R^{0}π_{∗}A_{𝔼,π}^{•} (induced by ad). The ϑ is an isomorphism such that the following diagram:

0 ⟶ R^{1}π_{∗}ω_{X/S} ⟶ R^{1}π_{∗}S(G_{E}) \underrightarrow{r̃es} R^{1}π_{∗}A_{E/S}^{0} ⟶ 0
r·id↓  ^{ϕ}↓  _{ϑ}↓
0 ⟶ R^{0}π_{∗}ω_{X/S}[1] ⟶ R^{0}π_{∗} ^{tr} A_{𝔼}^{•} ⟶^{res} R^{0}π_{∗}A_{𝔼,π}^{•} ⟶ 0

is commutaive. Thus ϕ must be an isomorphism. q.e.d.

Remark 3.17. The ϕ in i) of Theorem 3.7 is defined globally over M, but the one in ii) of Theorem 3.7 is defined only over an open set of M. More precisely, there is an affine covering {M_{i}}_{i∈I} of M such that on each C ×_{M} f^{−1}(M_{i}) we can choose a β_{i} : S(G_{E}) → ^{tr} A_{𝔼}^{−1} as the β in Proposition 3.16. Then, by using β_{i} and ii) of Theorem 3.7, we get the isomorphism ϕ_{i} : R^{1}π_{∗}S(G_{E}) → R^{0}π_{∗} ^{tr} A_{𝔼}^{•} = D_{S}^{≤1}(λ_{𝔼}) on f^{−1}(M_{i}). For another choice {β′_{i}}_{i∈I}, the map β_{i} − β′_{i} : S(G_{E}) → ω_{X/S} induces

ϕ_{i} − ϕ′_{i} : R^{1}π_{∗}S(G_{E}) → R^{0}π_{∗}ω_{X/S}[1] = O_{S}

on f^{−1}(M_{i}), which vanishes on R^{1}π_{∗}(G_{E}) (cf. Theorem 3.7 i, which is global on M), thus (ϕ_{i} − ϕ′_{i}) ∈ Ω_{M}^{1}(M_{i}). Similarly, on f^{−1}(M_{i} ∩ M_{j}), ϕ_{ij} := ϕ_{i} − ϕ_{j} induces ϕ̃_{ij} ∈ Ω_{M}^{1}(M_{i} ∩ M_{j}). Thus {ϕ̃_{ij}} defines a class in H^{1}(M, Ω_{M}^{1}), which is the obstruction of extending the isomorphism in ii) of Theorem 3.7 to the global M.

To conclude this section, we describe the connecting maps δ̃ and prove Lemma 2.4 (cf. Lemma 3.18).

Lemma 3.18. The map δ̃ induces the identity map on R^{1}π_{∗}(G_{E}). More precisely, δ̃ ∘ ι(•) = Sym^{2}((•) ⊗ 1).

Proof. It is known (see Proposition 4.2 of [16]) that the connecting map

δ̃: R^{1}π_{∗}(G_{E} ⊠ G_{E}(Δ)|_{Δ}) → R^{2}(π × π)_{∗}(G_{E} ⊠ G_{E})

is dual (under Serre duality) to the restriction map

r : (π × π)_{∗}(G_{E}^{∗} ⊠ G_{E}^{∗} ⊗ ω_{X/S} ⊠ ω_{X/S}) → π_{∗}(G_{E}^{∗} ⊗ G_{E}^{∗} ⊗ ω_{Δ/S}^{2}).

Working locally on S, we can assume that X is covered by affine open sets U and Ẋ. Let w ∈ ω_{X/S}(U̇) be a base of ω_{X/S} on U̇ := U ∩ Ẋ and w^{∗} ∈ T_{X/S}(U̇) be its dual base. Then, for any α ∈ G_{E}(U̇),

ι([α]) = [Sym^{2}(α ⊗ w) ⊗ w^{∗}] = \frac{1}{2}[(α ⊗ w + w ⊗ α) ⊗ w^{∗}].

We use the following identification:

(π × π)_{∗}(G_{E}^{∗} ⊠ G_{E}^{∗} ⊗ ω_{X/S} ⊠ ω_{X/S}) ≅ π_{∗}(G_{E}^{∗} ⊗ ω_{X/S}) ⊗ π_{∗}(G_{E}^{∗} ⊗ ω_{X/S}).

For any β_{i} ∈ π_{∗}(G_{E}^{∗} ⊗ ω_{X/S}) (i = 1, 2), let β_{i}|U̇ = s_{i} ⊗ w, where s_{i} ∈ G_{E}^{∗}(U̇) (i = 1, 2). Then r(β_{1} ⊗ β_{2})|U̇ = s_{1} ⊗ s_{2} ⊗ w ⊗ w. Thus

⟨δ̃(ι([α])), β_{1} ⊗ β_{2}⟩ = \frac{1}{2}[(s_{1}(α)s_{2}(w) + s_{1}(w)s_{2}(α)) · w]. (3.18)

By R^{1}π_{∗}ω_{X/S} = O_{S}, let [f · w] = 1 ∈ O_{S} (f ∈ O_{X}(U̇)), we have

⟨Sym^{2}([α] ⊗ [f · w]), β_{1} ⊗ β_{2}⟩ (3.19)

= \frac{1}{2}([s_{1}(α) · w][fs_{2}(w) · w] + [fs_{1}(w) · w][s_{2}(α) · w]).

Note that s_{i}(w) = β_{i}|U̇(w ⊗ w^{∗}) (i = 1, 2), we can see that s_{i}(w) = β_{i}(id)|U̇, where the (global) section id is the image of 1 under O_{X} → G_{E} ⊗ ω_{X/S}^{∗}. Thus s_{i}(w) ∈ O_{S} since β_{i} (i = 1, 2) are global sections of (G_{E} ⊗ ω_{X/S}^{∗})^{∗}. Then [s_{1}(α)s_{2}(w) · w] = [s_{1}(α) · w][fs_{2}(w) · w] and [s_{2}(α)s_{1}(w) · w] = [s_{2}(α) · w][fs_{1}(w) · w], which means that

⟨δ̃(ι([α])), •⟩ = ⟨Sym^{2}([α] ⊗ 1), •⟩.  q.e.d.

4. Generalization to the singular case

Let π : X̃ → S̃ be a proper morphism of relative dimension 1 between smooth varieties in characteristic 0 such that each fiber has at most ordinary double points as singularities. Let f̃: S̃ → M̃ be a smooth morphism where M̃ is a smooth variety. Let B = M̃ \ M and W = f̃^{−1}(B) such that D = π^{−1}(W) consists precisely of singular fibres. As before let ω_{X̃/S̃} be the relative dualizing sheaf (which is locally free as is well-known). Let TM̃(B) ⊂ TM̃ be the subalgebra of vector fields that preserve B (cf. [2], Section 6).

Notation 4.1. In the notation of Section 3, define the following:

Tf̃(log D) = d(f̃∘ π)^{−1}((f̃∘ π)^{−1}TM̃(B)) ⊂ TX̃;
T_{π}(log D) = Tf̃(log D) ∩ dπ^{−1}(π^{−1}TS̃) ⊂ TX̃;
TS̃(log W) = dπ(T_{π}(log D)) ⊂ TS̃;
T_{X̃/S̃}(log D) = T_{X̃/S̃} ∩ T_{π}(log D).

Notation 4.2. Let E be a vector bundle on X̃. Define the following:

A_{E,π}(log D) = ϵ^{−1}T_{π}(log D) ⊂ A_{E};
A_{E/S̃}(log D) = ϵ^{−1}T_{X̃/S̃}(log D) ⊂ A_{E};
A_{λ_{E}}(log W) = ϵ^{−1}TS̃(log W) ⊂ A_{λ_{E}},

where the “ϵ”s denote symbol maps.

The ^{tr}A_{E}^{−1} in Section 3 admits a generalization ^{tr}A_{E}^{−1}(log D) (cf. [19], p. 593) such that

(4.1) 0 → ω_{X̃/S̃} → ^{tr}A_{E}^{−1}(log D) → A_{E/S̃}(log D) → 0,

is exact. Furthermore with ^{tr}A_{E}^{•} in Section 3 replaced by ^{tr}A_{E}^{•}(log D) (with A_{E,π}(log D) in degree 0), one has:

Proposition 4.3 (cf. [19]). There is a canonical isomorphism

R^{0}π_{∗}^{tr} A_{E}^{•}(log D) ≅ A_{λ_{E}}(log W)

that extends Proposition 3.1.

Now go back to moduli situation, note T_{X̃/S̃}(log D) = ω_{X̃/S̃}^{−1}. Recall Notation 2.10 and Kodaira–Spencer map (cf. [20], Remark 3.2.7)

KS : f̃^{∗}TM̃(B) → R^{1}π_{∗}T_{X̃/S̃}(log D). (4.2)

As in the situation of smooth curves, we have:

Lemma 4.4. The morphism f̃: S̃ → M̃ is smooth and

T_{S̃/M̃} = R^{1}π_{∗}End^{0}(E).

Proof. When C is irreducible, its fibre f̃^{−1}([C]) is the moduli space of stable bundles with fixed determinant N |_{C}. When C is reducible, f̃^{−1}([C]) have a few disjoint irreducible components and each component consists of bundles with a fixed determinant that coincides with N |_{C} outside the node of C (cf. [18]). q.e.d.

336 X. SUN & I-H. TSAI

Proposition 4.5.

i) Assume the Kodaira–Spencer map KS (4.2) is injective. Then R^{0}π_{∗}(End(E)^{−1} → A_{E,π}(log D)) → R^{0}π_{∗}(End(E) → A_{E,π}(log D)) → 0

is canonically isomorphic to

D_{S̃/M̃}^{≤1}(λ_{E}) → T_{S̃/M̃} → 0.

ii) Assume that KS (4.2) is an isomorphism. Then

R^{1}π_{∗}A_{E/S̃}^{0}(log D) ≅ TS̃(log W)

canonically, where A_{E/S̃}^{0}(log D) = A_{E/S̃}(log D)/OX̃. The following is left to the reader: Lemma 4.6.

(A_{E/S̃}(log D) → A_{E,π}(log D)) ≅_{q.i.} (T_{X̃/S̃}(log D) → T_{π}(log D))
≅_{q.i.} π^{−1}TS̃(log W)

as quasi-isomorphisms.

Proof of Proposition 4.5.

i) From the exact sequence 0 → (End(E)^{−1} → A_{E,π}(log D)) →^{tr} A_{E}^{•}(log D) → T_{X̃/S̃}(log D)[1] → 0, and passage to cohomology

0 → R^{0}π_{∗}(End(E)^{−1} → A_{E,π}(log D)) → A_{λ_{E}}(log W) ⟶^{ϵ}

R^{0}π_{∗}T_{X̃/S̃}(log D)[1] → ⋯,

one has, via the injectivity of KS, that ϵ^{−1}(0) has symbolic part in T_{S̃/M̃}. This gives one of the isomorphisms in i) (the one with D_{S̃/M̃}^{≤1}(λ_{E})). Further, by

0 → ω_{X̃/S̃}[1] → (End(E)^{−1} → A_{E,π}(log D))

→ (End(E) → A_{E,π}(log D)) → 0

and R^{0}π_{∗}ω_{X̃/S̃}[1] ≅ OS̃ it follows

R^{0}π_{∗}(End(E)^{−1} → A_{E,π}(log D)) → R^{0}π_{∗}(End(E) → A_{E,π}(log D)) is nothing but the symbol map, completing the asserted isomorphisms.

ii) Write B^{•}(log D) for (End(E) → A_{E,π}(log D)) and A_{E,π}^{•}(log D) for L.H.S. of Lemma 4.6. The following exact sequence:

(4.3) 0 → B^{•}(log D) → A_{E,π}^{•}(log D) → T_{X̃/S̃}(log D)[1] → 0

projects to

0 → End^{0}(E)[1] → A_{E/S̃}^{0}(log D)[1] → T_{X̃/S̃}(log D)[1] → 0. (4.4)

Computing direct images of (4.3) and (4.4), by i) just proved and Lemma 4.6, yields that R^{0}π_{∗} of (4.4) should be isomorphic to

0 → T_{S̃/M̃} → TS̃(log W) → f̃^{∗}TM̃(B) → 0,

implying the assertion. q.e.d.

Remark 4.7.

i) R^{1}π_{∗}(G_{E}) ≅ R^{0}π_{∗}(End(𝔼)^{−1} → A_{𝔼,π}(log D)) ≅ D_{S̃/M̃}^{≤1}(λ_{𝔼}) holds, cf. Theorem 3.7 and Remark 3.8.

ii) For the family C̃ → M̃ there is a Kodaira–Spencer map (cf. [20], 3.1–2)

ρ_{b} : T_{M̃,b} → Ext_{OC̃_{b}}^{1} (ΩC̃_{b}, OC̃_{b}), b ∈ M̃. (4.5)

The family C̃ → M̃ is a local universal family if ρ_{b} is an isomorphism at each b ∈ M̃. If C̃ → M̃ is a local universal family (cf. [20], Theorem 3.1.5 for the existence of such a family), then KS (4.2) is an isomorphism (cf. [20], Theorem 3.2.6).

Combining the above with the second half of Section 3, we are now ready to generalize Theorem 3.7 in the context of log geometry.

Theorem 4.8. Suppose KS in (4.2) is an isomorphism (cf. Remark 4.7 ii). Then, over any affine open set U_{i} of M̃, there is an isomorphism

ϕ_{i} : R^{1}π_{∗}S(G_{E}) ≅ D≤1 ; S̃(λ_{𝔼})(log W).

Proof. Note the above assumption for KS is, via ii) of Proposition 4.5, in correspondence to KS_{S} in Theorem 3.7. It follows that all key ingredients in proof of Theorem 3.7 admit corresponding counterparts for log geometry, such as Proposition 4.3 and Proposition 4.5. Thus the generalization of Theorem 3.7 in log context is immediate. q.e.d.

To complete this paper, we prove the coherence of f_{T ∗}Θ^{k} (for definitions of f_{T} and f_{Z}, see Notation 2.10), for which our proof needs a result on the density of locally free sheaves.

Proposition 4.9. The fibre of f_{Z} : Z → M̃ at any point of B = M̃ \ M has a dense open set of locally free sheaves.

Proof. Let X_{0} be a fibre of C → M̃ at 0 ∈ B. If X_{0} is reducible, the lemma is known (see Theorem 1.6 and Lemma 2.2 of [18]). Thus we assume that X_{0} is irreducible. Let U_{0} be the moduli space of semistable sheaves of rank r and degree d (without fixed determinant) on X_{0}. We need to show that Z_{0} := f_{Z}^{−1}(0) (⊂ U) contains a dense set of locally free sheaves with the fixed determinant N_{0}. Let J(X_{0}) be the Jacobian of X_{0}, which consists of line bundles of degree 0 (thus noncompact for the singular curve X_{0}). Then we have a morphism

ϕ_{0} : Z_{0} × J(X_{0}) → U_{0}, where ϕ_{0}(E, L) = E ⊗ L.

Now we prove that ϕ_{0} has fibre dimension at most 1, namely, for any [F_{0}] ∈ U_{0}, the fibre

ϕ_{0}^{−1}([F_{0}]) = {(F, L)|F ⊗ L = F_{0}} ⊂ Z_{0} × J(X_{0})

has at most dimension 1 (for simplicity, we assume that X_{0} has only one node x_{0}). One can check that for any [F] ∈ Z_{0} it satisfies

\frac{∧^{r}F}{torsion} ⊂ N_{0} and m_{x_{0}}^{r(F)}N_{0} ⊂ \frac{∧^{r}F}{torsion}

where m_{x_{0}} is the ideal sheaf of the node x_{0} ∈ X_{0} and r(F) = d − deg(\frac{∧^{r}F}{torsion}). Let ρ : X̃_{0} → X_{0} be the normalization and ρ^{−1}(x_{0}) = {x_{1}, x_{2}}. Then the above condition implies that

\frac{ρ^{∗}(∧^{r}F)}{torsion} = ρ^{∗}N_{0}(−n_{1}x_{1} − n_{2}x_{2}), n_{1} + n_{2} ≤ 2r(F), n_{1} ≥ 0, n_{2} ≥ 0,

where r(F) = d − deg(\frac{∧^{r}F_{0}}{torsion}) = r(F_{0}) since F ⊗ L = F_{0} and thus

\frac{∧^{r}F_{0}}{torsion} = \frac{∧^{r}F}{torsion} ⊗ L^{r}.

Thus, for any (F, L) ∈ ϕ_{0}^{−1}(F_{0}), L has to satisfy

(ρ^{∗}L)^{r} = (ρ^{∗}N_{0})^{−1}(n_{1}x_{1} + n_{2}x_{2}) ⊗ \frac{ρ^{∗} ∧^{r} F_{0}}{torsion}

which is a finite set since there are only finitely many choices of (n_{1}, n_{2}). The pullback map ρ^{∗} : J(X_{0}) → J(X̃_{0}) has 1-dimensional kernel. Thus dim(ϕ_{0}^{−1}([F_{0}])) ≤ 1. We shall now prove the density of locally free sheaves.

Let Z_{0}^{i} be an irreducible component of Z_{0}, then

(4.6) dim(Z_{0}^{i} × J(X_{0})) ≥ dim(Z_{η} × J(X_{0})) = dim(U_{0}).

If Z_{0}^{i} contains no locally free sheaf, then ϕ(Z_{0}^{i} × J(X_{0})) falls into the subvariety U_{0}^{n} ⊂ U_{0} of nonlocally free sheaves. U_{0}^{n} has a dense open set U_{0}^{n}(1) consisting of torsion-free sheaves F of the following type, said to be type 1. Namely:

F ⊗ Ô_{X_{0},x_{0}} = Ô_{X_{0},x_{0}}^{(r−1)} ⊕ m̂_{x_{0}}.

If ϕ((F, L)) = F_{0} ∈ U_{0}^{n}(1), then F is also of type 1 (tensoring a line bundle does not change its type). By Remark 8.1 of [15],

L_{0} = \frac{∧^{r}F}{torsion}

is a torsion-free (but nonlocally free) sheaf of degree d. But L_{0} ⊂ N_{0}, thus L_{0} = N_{0} since they have the same degree, a contradiction with that L_{0} is not locally free. Thus ϕ(Z_{0}^{i} × J(X_{0})) falls into a subvariety of codimension at least two, which contradicts the dimension of fibres and (4.6). q.e.d.

Theorem 4.10.

i) f_{T ∗}Θ^{k} is coherent.

ii) f_{T ∗}Θ^{k} = f̃_{∗}Θ^{k} if either g ≥ 3 or r ≥ 3.

Proof. Let ι : Z̃ → Z be the normalisation of Z and Θ̃ = ι^{∗}(Θ). Write fZ̃ : Z̃ → M̃ and f_{ι^{−1}(T)} : ι^{−1}(T) → M̃. Then ι^{−1}(T) ≅ T since T is normal, and F := f_{T ∗}Θ^{k} ≅ f_{ι^{−1}(T)∗}(Θ̃^{k}). On the other hand, since each fibre of f_{Z} : Z → M̃ contains a dense open set of locally free sheaves, we have codim(Z \ T) ≥ 2. Thus Z̃ \ ι^{−1}(T) = ι^{−1}(Z \ T) has codimension at least 2 since ι : Z̃ → Z is a finite map. By Hartogs type extension theorem,

F ≅ f_{ι^{−1}(T)∗}(Θ̃^{k}) ≅ f_{Z̃∗}(Θ̃^{k}),

which is coherent, hence i). The claim ii) that f_{T ∗}Θ^{k} = f̃_{∗}Θ^{k} follows also from the Hartogs type theorem because T is normal and T \ S̃ is of codimension at least 2 when g > 2 or r > 2 (cf. [10]). q.e.d.

Finally, we prove the following lemma, though not strictly needed for this paper, which gives the relationships of S(G_{E}), G_{E^{∗}} and A_{E^{∗}/S}. We remark that the morphism r̃es in Proposition 3.13 can be induced by pairings in this lemma.

Lemma 4.11. There are canonical isomorphisms:

i) S(G_{E})/ω_{X/S} ≅ (G_{E^{∗}})^{∗}, ii) (G_{E})^{∗} ≅ A_{E^{∗}/S}^{0} where A_{E^{∗}/S}^{0} = A_{E^{∗}/S}/O_{X}.

Proof.

i) One constructs a nondegenerate pairing

G_{E^{∗}} × S(G_{E})/ω_{X/S} → O_{X}

using local description (3.7) (also cf. [2]). We define

⟨s_{1}, s_{2}⟩ := χν + \frac{1}{2} ∑_{a}(ν_{a} + μ_{a})Trace(J_{a} · ^{t}B) (4.7)

for s_{1} = (0, B, ν) and

s_{2} = ( \begin{matrix}χ ∑_{a}(0, J_{a}, 0) ⊗ (0, J_{a}, 0) + ∑_{a} μ_{a}(0, J_{a}, 0) ⊗ (0, 0, 1) \\ + ∑_{a} ν_{a}(0, 0, 1) ⊗ (0, J_{a}, 0) + w(0, 0, 1) ⊗ (0, 0, 1) \end{matrix}) ⊗ ∂_{t}.

One sees that ω_{X/S} is contained in the kernel of the pairing (4.7). The pairing is obviously invariant under coordinates change y(t). If the trivialization of E is changed by a gauge g, then E^{∗} is changed by a gauge (^{t}g)^{−1}. Thus the verification of (4.7) being g-invariant is easily reduced to an identity

∑_{a} Tr(−J_{a}g^{−1}g′) · Tr(J_{a} ·^{t}B) = Tr(B(^{t}g)((^{t}g)^{−1})′). (4.8)

The left-hand side of (4.8) equals Tr(−g^{−1}g′ ·^{t}B) (B being traceless). The right-hand side of (4.8), after transposition, is

Tr((g^{−1})′g ·^{t}B) = Tr((−g^{−1}g′g^{−1})g ·^{t} B).

ii) Define a nondegenerate pairing

G_{E} × A_{E^{∗}/S}^{0} → O_{X} (4.9) by ⟨(0, B_{1}, ν), (χ, B_{2})⟩ = νχ + Trace(B_{1} ·^{t} B_{2}). We check that it is independent of choices of coordinates and gauges.

(4.10) (0, B_{1}, ν) := s_{1} ⟶^{g} (0, gB_{1}g^{−1}, Tr(−B_{1}g^{−1}g′) + ν);
(χ, B_{2}) := s_{2} ⟶^{^{t}g^{−1}} (χ, −χ((^{t}g^{−1})′ ·^{t}g) + (^{t}g^{−1})B_{2} ·^{t} g);
s_{1} ⟶^{y(t)} (0, B_{1}, νy′);
s_{2} ⟶^{y(t)} (χy′^{−1}, B_{2}).

The y(t)-change is obvious. For g-change, we have

(4.11) χν + Tr(B_{1} ·^{t} B_{2}) ⟶χν + χTr(−B_{1}g^{−1}g′)

+ Tr(gB_{1}g^{−1} ·^{t} ( − χ((^{t}g^{−1})′ ·^{t}g))

+ Tr(gB_{1}g^{−1} ·^{t} (^{t}g^{−1}B_{2} ·^{t} g)).

The first (resp. second) term in the last line equals

(4.12) −χTr(gB_{1}g^{−1} · g(g^{−1})′) = −χTr(gB_{1}g^{−1} · g(−g^{−1}g′g^{−1}))
= χTr(g(B_{1}g^{−1}g′)g^{−1})
= χTr(B_{1}g^{−1}g′)
(resp. Tr(gB_{1} ·^{t} B_{2}g^{−1}) = Tr(B_{1} ·^{t} B_{2})).

It follows from (4.11) and (4.12) that the pairing χν + Tr(B_{1} ^{t}B_{2}) in (4.9) is globally defined. q.e.d.

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Institute of Mathematics

AMSS, Chinese Academy of Sciences

China

E-mail address: xsun@mail2.math.ac.cn

and

Department of Mathematics The University of Hong Kong Pokfulam Road, Hong Kong E-mail address: xsun@maths.hku.hk

Department of Mathematics National Taiwan University, Taipei

Taiwan

E-mail address: ihtsai@math.ntu.edu.tw