j. differential geometry

55 (2000) 193-227

MODULI OF SHEAVES ON SURFACES AND ACTION OF THE OSCILLATOR ALGEBRA

VLADIMIR BARANOVSKY

Abstract

This paper gives a generalization of some results on Hilbert schemes of points on surfaces. Let M^{G}(r, n) (resp. M^{U} (r, n)) be the Gieseker (resp. Uhlenbeck) compactification of the moduli spaces of stable bundles on a smooth projective surface. We show that, for surfaces satisfying some technical condition:

(a) The natural map M^{G}(r, n) → M^{U} (r, n) generalizing the Hilbert-Chow morphism from the Hilbert scheme of n points on S to the n-th symmetric power, is strictly semi-small in the sense of Goresky-MacPherson with respect to some stratification.

(b) Let P_{t}(X) be the Intersection Homology Poincare polynomial of X. Generalizing the computation due to Gottsche and Sorgel we

prove that the ratio \frac{∑_{n} q^{n}P_{t}(M^{G}(r,n))}{∑_{n} q^{n}P_{t}(M^{U} (r,n))} is a character of a certain Heisenberg-type algebra.

(c) Generalizing results of Nakajima we show how to obtain the action of the Heisenberg algebra on the cohomology using correspondences.

Introduction

Let S be a smooth complex projective surface and Hilb^{n}(S) the Hilbert scheme of length n zero-dimension subschemes of S. It is known (cf. [10]) that Hilb^{n}(S) is a smooth projective variety of dimension 2n. The structure of the cohomology ring of Hilb^{n}(S) for a fixed n is rather difficult to understand. However, when we consider the direct sum ⨁_{n≥0} H^{∗}(Hilb^{n}(S)) of the cohomology groups (always with complex

Received May 23, 1999.

coefficients in this paper) over all possible n, the picture becomes more comprehensible.

For any complex smooth algebraic variety X of dimension d denote by P_{t}(X) the shifted Poincaré polynomial ∑_{i=0}^{d} t^{i−d}·dim_{ℂ} H^{i}(X). It was shown by Göttsche [Gö1] that, for any smooth quasi-projective surface S one has an identity

∑_{n≥0} q^{n}P_{t}(Hilb^{n}(S)) = ∏_{l=1}^{∞} \frac{(1 + t^{−1}q^{l})^{b_{1}(S)}(1 + tq^{l})^{b_{3}(S)}}{(1 − t^{−2}q^{l})^{b_{0}(S)}(1 − q^{l})^{b_{2}(S)}(1 − t^{2}q^{l})^{b_{4}(S)}} ,

where b_{i}(S) are the Betti numbers of S.

Vafa and Witten [32] have noticed that the right hand side of the formula above is an irreducible character of the oscillator algebra (or Heisenberg/Clifford algebra) H defined for a smooth projective surface S as follows:

(a) H is generated by elements p_{i}^{α}, α ∈ H^{∗}(S), i ∈ ℤ \ 0.

(b) The generators satisfy [p_{i}^{α}, p_{j}^{β}] = ⟨α, β⟩i · δ_{i+j,0},

where the commutator is understood in the graded sense and ⟨·, ·⟩ is the intersection form on S.

One could expect to construct an actual action of the oscillator algebra on the cohomology groups, since one of the standard realizations of the Fock representation is in the space of all symmetric polynomials in infinitely many variables. This realization appears from the cohomology of symmetric powers Sym^{n}(S) related to Hilbert schemes of points via the Hilbert-Chow morphism Hilb^{n}(S) → Sym^{n}(S).

And indeed, Nakajima and Grojnowski (cf. [27], [15]) have constructed the expected action of H on ⨁_{n≥0} H^{∗}(Hilb^{n}(S)) using some ex-

plicit cycles in the products of Hilbert schemes.

The Hilbert scheme can be viewed as a moduli space of rank one sheaves on S with trivial determinant and c_{2} = n. It was conjectured by Vafa and Witten in [32] (see also [27], [15]) that there should be a higher rank extension of the results above. This paper provides such a generalization (cf. Theorem 4.1). The key obsevation which stands behind our arguments is that in certain cases a non-compact moduli space M may admit two different natural compactifications M_{1}, M_{2} and a map M_{1} → M_{2} which (partially) resolves the singularities of M_{2} and restricts to identity on the copies of M in M_{1} and M_{2}.

The list of examples includes moduli of maps of curves to flag varieties (M_{1} and M_{2} are Laumon and Drinfeld compactifications, respectively, cf [20]), instantons on ALE spaces (quiver variety and the Uhlenbeck-type compactification cf. [26]), moduli of abelian varieties (Voronoi and Satake compactifications) and, finally, moduli of stable bundles on surfaces (Gieseker and Uhlenbeck compactifications). Thus, the Hilbert scheme is replaced by the Gieseker moduli space M^{G}(r, n) of stable torsion-free sheaves and the role of the symmetric power is played by the Uhlenbeck compactification M^{U}(r, n).

The first key result of this paper is that the fibers of the natural map M^{G}(r, n) → M^{U}(r, n) are irreducible of expected dimension. (This generalization of the results by Briançon and Iarrobino (cf. [5], [18]) was also proved independently by Ellingsrud and Lehn, cf. [9]).

This leads to the second important theorem which says that, in the coprime case, the natural morphism M^{G}(r, n) → M^{U}(r, n) is strictly semi-small in the sense of Goresky-MacPherson (if a certain technical condition is satisfied: the surface S is allowed, for instance, to be rational, birationally ruled, K3 or abelian). This result was originally conjectured by V. Ginzburg and the corresponding statement is true for some of the moduli spaces mentioned above. We proceed further to give a natural generalization of the rank one correspondences yielding an action of the oscillator algebra on ⨁_{n≥0} H^{∗}(M^{G}(r, n)). We show

how to extend Nakajima’s proof of commutation relations to the higher ranks case. Some parts of the proof require a more detailed analysis of the geometry of moduli spaces, than in the case of Hilbert schemes (see Sections 5 and 6).

It was communicated to us by G. Moore that L^{2}-cohomology groups of instanton moduli spaces have recently appeared in string theory. Since L^{2}-cohomology of a non-compact moduli space coincides in several interesting cases with intersection homology of its particular compactification, this provides another motivation for studying these homology groups. This work can be regarded as a first step in application of intersection homology to moduli spaces of sheaves on surfaces.

Acknowledgements. I would like to thank Victor Ginzburg whose conjecture has originally motivated this work. I am grateful to Lothar Göttsche, Stein Arild Strømme, Zhenbo Qin, Gregory Moore and Hiraku Nakajima for the useful discussions, advices and remarks.

A part of this work was carried out during my stay at the Max Planck Institut für Mathematik in Bonn. I want to thank all the people involved, particularly Don Zagier, for the invitation and the excellent research conditions.

1. The moduli spaces

Let S be a smooth complex projective surface. We are interested in the study of the moduli space of rank r vector bundles on S. It is known, cf. [22], that all bundles cannot be parametrized by points of an algebraic variety, because of the problems arising from vector bundles with large automorphism groups. Instead one restricts attention only to “good” bundles, i.e those which are Gieseker-stable (cf. [17]) with respect to a fixed ample line bundle H. Moreover, in this paper we will only consider the situation when all other notions of “good” bundles, Gieseker semistability and Mumford (semi)stability, coincide with Gieseker stability (see next section).

Thus, we consider the moduli space N(r, n) of Gieseker H-stable vector bundles E of rank r with fixed determinant L and c_{2}(E) = n. The line bundles L and H will be fixed throughout this paper. It is known, (cf. for example [17]), that N(r, n) is empty unless the following Bogomolov-Miyaoka inequality is satisfied

n − \frac{r − 1}{2r}c_{1}(L)^{2} ≥ 0 (1)

The moduli space N(r, n) is non-compact and it can be compactified in two different ways. The first compactification is the Gieseker moduli space M^{G}(r, n) of all H-stable torsion-free sheaves E of rank r with det(E) = L and c_{2}(E) = n (cf. [22]).

The second compactification is the Uhlenbeck moduli space M^{U}(r, n) which can be described as follows. Denote by Sym^{k}S the k-the symmetric power of S, and start with the disjoint union ∐_{s=0}^{∞} N(r, n − s) × Sym^{s}S (note that this union is in fact finite by (1)). A theorem due to Uhlenbeck [31] says that any sequence of points in one piece of this disjoint union has a subsequence that converges (in some sense) to a point in another piece. This endows ⋃_{s=0}^{∞} N(r, n − s) × Sym^{s}S with a topology of a compact space. One can show (cf. [17, Chapter 8]) that this topological space can be endowed with a structure of an algebraic variety. This variety is called the Uhlenbeck compactification M^{U}(r, n) of N(r, n).

In general, the two compactifications above may be quite difficult to investigate: N(r, n) may not be dense in M^{G}(r, n) or M^{U}(r, n), there may be components of dimension higher than expected, etc. Hence we impose a technical condition which ensures that the moduli spaces are well-behaved.

The Main Technical Condition

(a) The integers r and d := c_{1}(L) · c_{1}(H) are coprime.

(b) Either the canonical bundle K_{S} is trivial or c_{1}(K_{S}) · c_{1}(H) < 0.

The main reason for imposing (a) and (b) is that they guarantee (cf. [17]) that for any n the moduli space M^{G}(r, n) is either empty or smooth of expected dimension

dim M^{G}(r, n) = 2rn − (r − 1)(c_{1}(L))^{2} − (r^{2} − 1)χ(O_{S}) + h^{1}(O_{S}). (2)

Condition (b) is automatic when (−K) can be represented by an effective curve. Moreover, since N(r, n) is an open subset of M^{G}(r, n) (loc. cit.), it is also smooth.

Since r and d are assumed to be coprime, the two possible notions of stability (Mumford-Takemoto and Gieseker) coincide.

Examples. If we require that K_{S} is trivial or effective then S can be a Del Pezzo surface, K3 or abelian surface. If we make a special choice of H as above, the list of examples will extend to all rational surfaces and birationally ruled surfaces.

It is known (e.g., [8, 10.3.4]) that for some elliptic surfaces or surfaces of general type the condition (2) may fail.

2. Gieseker to Uhlenbeck

By definition the Hilbert scheme Hilb^{n}(S) parametrizes Artin subschemes ξ ∈ S of length n. Every such subscheme is determined by its ideal sheaf J_{ξ} ⊂ O_{S}. The sheaves J_{ξ} which arise in this way can be characterized as torsion-free sheaves of rank one with trivial determinant and c_{2} = n.

Thus, M^{G}(r, n) is a higher rank analogue of Hilb^{n}(S). We want to generalize the natural map a_{1} : Hilb^{n}(S) → Sym^{n}(S) given by

a_{1} : J_{ξ} ↦ ∑_{x_{i}∈Supp ξ} mult_{x_{i}}(ξ) · x_{i}

where the formal sum on the left hand side is viewed as an element of Sym^{n}(S) (the rigorous construction of a_{1} uses the notion of “linear determinant”, see [17]).

To define the generalization a_{r} : M^{G}(r, n) → M^{U}(r, n) of the map a_{1}, for every torsion-free sheaf F ∈ M^{G}(r, n) we consider the double dual F^{∗∗}. Since F^{∗∗} is reflexive (i.e., coincides with its own double dual) and dim S = 2, we can apply Lemma 1.1.10 in [28] and conclude that F^{∗∗} is locally free.

Now consider the short exact sequence:

0 → F → F^{∗∗} → A_{F} → 0.

The quotient sheaf A_{F} = F^{∗∗}/F is an Artin sheaf of a certain length l_{F} supported by finitely many points with multiplicities which add up to l_{F}. We define a_{r}(F) ∈ M^{U}(r, n) by

a_{r} : F ↦ (F^{∗∗}, ∑_{x_{i}∈Supp A_{F}} mult_{x_{i}}(A_{F}) · x_{i})

where the image is a point in N(r, n − l_{F}) × Sym^{l_{F}} (S) ⊂ M^{U}(r, n).

Remark. Another consequence of r and d = c_{1}(H) · c_{1}(L) being coprime is that any semistable sheaf is necessarily stable. This ensures that F^{∗∗} is stable if F is, hence the definition above makes sense. If r and d are not coprime, the space M^{G}(r, n) parametrizes only S-equivalence classes (cf. [22]) of semistable sheaves and one has to be more careful to make sure that a_{r} is well-defined on such classes (cf. [21]).

The algebraic structure on M^{U}(r, n) is defined in such a way that the map a_{r} is automatically algebraic (cf. [17, Chapter 8] and also the orginal paper by J. Li [21]).

Now we describe the fiber of a_{r} over an arbitrary point p = (E, ∑_{i} m_{i}x_{i}) in M^{U}(r, n). Here m_{i} are positive integers and the points x_{i} are pairwise distinct. The fiber a_{r}^{−1}(p) parametrizes all the quotients E → A → 0 where the sheaf A is supported at the points x_{i} with prescribed multiplicities m_{i}. Since the question is local, the fiber does not depend on E and points x_{i} but only on r and m_{i}. To be more precise, let Quot(r, n) denote the punctual Quot scheme of all quotients O^{⊕r} → A of fixed length n which are supported at a fixed point x. This scheme depends only on completion Ô_{x} of the local ring O_{x} of x, hence different points x lead to isomorphic Quot schemes. Since every vector bundle is locally trivial by definition, the following statement is immediate

Proposition 2.1. The fiber a_{r}^{−1}(p) over the point p = (E, ∑_{i} m_{i}x_{i}) is isomorphic to the product of punctual Quot schemes ∏_{i} Quot(r, m_{i}).

The proposition above motivates the following theorem (cf. [2] and [9]):

Theorem 2.2. The punctual Quot scheme Quot(r, n) defined above, is irreducible of dimension rn − 1.

Proof. See Appendix. q.e.d.

The first application of Theorem 2.2 is to show that, under the Main Technical Condition, the moduli spaces M^{G}(r, n) and M^{U}(r, n) are well-behaved.

Theorem 2.3. Assume that the Main Technical Condition of Section 1 is satisfied. Then the following statements hold:

(a) N(r, n) is dense in M^{G}(r, n);

(b) N(r, n) is dense in M^{U}(r, n);

(c) Any irreducible component of N(r, n − s) × Sym^{s}(S) intersects the closure of a unique component of N(r, n) (and hence by (b) is contained in it).

Proof. Note that (a) implies (b) since a_{r} is surjective and one-to-one on the copies of N(r, n) in M^{G}(r, n) and M^{U}(r, n), respectively.

To prove (a) suppose that N(r, n) is not dense in M^{G}(r, n). Then there exists a component of M^{G}(r, n) such that generic point of it corresponds to a non-locally free sheaf. This means that the image of this component in M^{U}(r, n) is a subset of ⋃_{i≥1} N(r, n − s) × Sym^{s}(S). Since all the components of N(r, n) have expected dimension (and not more than that), Theorem 2.2 above implies that the dimension of a_{r}^{−1}(⋃_{i≥1} N(r, n − s) × Sym^{s}(S)) is strictly less than the expected dimension of M^{G}(r, n) which is impossible.

Finally, if (c) were false, some irreducible component X of N(r, n − s) × Sym^{s}(S) would intersect the closures of at least two different components of N(r, n). Since M^{G}(r, n) is smooth, these would mean that X would belong to the image of two different connected components of M^{G}(r, n). But this is impossible since all fibers of a_{r} are irreducible.

q.e.d.

Corollary 2.4. M^{U}(r, n) is a disjoint union of the closures of irreducible components of N(r, n).

3. Stratifications and semi-small maps

From now on we will assume that the Main Technical Condition on the pair (S, H) is satisfied.

Recall briefly the results on symmetric products and Hilbert schemes. Let μ be a partition of n. Any such μ can be represented either by a

non-increasing sequence μ = (μ_{1} ≥ μ_{2} ≥ … ≥ μ_{m} > 0) with ∑ μ_{i} = n or in the form 1^{m_{1}}2^{m_{2}} … n^{m_{n}}, where m_{i} is the number of parts μ which are equal to i (hence m_{i} ≥ 0). Note that m_{1} + m_{2} + … + m_{n} is equal to the number m of non-zero parts of μ.

The symmetric product Sym^{n}(S) has a natural stratification by locally closed strata labeled by partitions of n.

The stratum Sym_{μ}^{n}(S) is the set of formal sums of the type ∑ μ_{i}x_{i} (viewed as elements of Sym^{n}S) where x_{i} are pairwise distinct. Note that Sym_{(1,... ,1)}^{n} is a dense open subset of Sym^{n}(S) and in general Sym_{μ}^{n}(S) is isomorphic to a dense open subset of Sym^{m_{1}}(S) × … × Sym^{m_{n}}(S). A generic element (y_{1}, …, y_{n}) in the latter product corresponds to the point y_{1} + 2y_{2} + … + ny_{n} in Sym_{μ}^{n}(S).

Let π : Z → Y be a proper projective morphism of algebraic varieties. Suppose that Y decomposes into a finite number of locally closed strata: Y = ⋃_{μ} Y_{μ} and choose an arbitrary point y_{μ} ∈ Y_{μ}. Assume that the restriction π : π^{−1}(Y_{μ}) → Y_{μ} is a topological fiber bundle with fiber π^{−1}(y_{μ}).

Definition (cf. [4] or [7, Chapter 8]). The map π is called strictly semi-small if it satisfies

2 dim π^{−1}(y_{μ}) = codim Y_{μ}

for any stratum Y_{μ}.

The following proposition is an immediate consequence of results of Briançon and Iarrobino (cf. [5], [18]).

Proposition 3.1 (cf. [14]).

The morphism a_{1} : Hilb^{n}(S) → Sym^{n}(S) is strictly semi-small, with respect to the stratification given by Sym_{μ}^{n}(S).

We will give a generalization of this proposition to the case of arbitrary rank. Exactly as in [14] this will lead to some formulas for Poincaré polynomials. These formulas will be later interpreted in representationtheoretic terms.

The first step is to stratify M^{U}(r, n) appropriately since the natural strata coming from the definition of M^{U}(r, n) are too big. The new finer strata will be labeled by pairs (s, μ) where s < n is a non-negative integer such that N(r, n − s) is non-empty, and μ = (μ_{1} ≥ μ_{2} ≥ … ≥ μ_{m} ≥ 0) is a partition of s. For each pair (s, μ) we consider M_{s,μ}^{U}(r, n) = N(r, n − s) × Sym_{μ}^{n}(S) which is naturally a locally closed subset of M^{U}(r, n). Of course, this is nothing but a ”common refinement” of the natural partition of M^{U}(r, n) and the above partition above of the symmetric product.

Part (b) of the next theorem was originally conjectured by V. Ginzburg. It provides a starting point for our generalization of Nakajima’s construction.

Proposition 3.2. Let s and μ = (μ_{1} ≥ μ_{2} ≥ … ≥ μ_{m} > 0) be as above and let M_{s,μ}^{U}(r, n) be the stratum associated to the pair (s, μ). Then:

(a) For any point x_{s,μ} ∈ M_{s,μ}^{U}(r, n) the dimension of the fiber a_{r}^{−1}(x_{s,μ}) is equal to (rs − m) where m is the number of non-zero parts of μ.

(b) The morphism M^{G}(r, n) → M^{U}(r, n) is strictly semi-small with respect ot the stratification given by M_{s,μ}^{U}(r, n).

Proof. To prove (a) note that by Proposition 2.1 the fiber is isomorphic to ∏ Quot(r, m_{i}) and by Theorem 2.2 its dimension is equal to ∑_{i=1}^{m}(rμ_{i} − 1) = r(∑_{i=1}^{m} μ_{i}) − m = rs − m.

To prove (b) we have to show that codim M_{s,μ}^{U} = 2dim a_{r}^{−1}(x_{s,μ}) = 2(rs − m). In fact, codim M_{s,μ}^{U}(r, n) = dim N(r, n) − dim N(r, n − s) − dim Sym_{μ}^{n}(S) = 2rs − 2m = 2(rs − m). q.e.d.

The semi-smallness result will allow us to relate homological invariants of M^{G}(r, n) and M^{U}(r, n). Recall (cf. [3]) that for any algebraic variety X, there exists a remarkable complex of sheaves IC_{X} (intersection cohomology complex) such that its cohomology groups IH^{∗}(X) = H^{∗}(X, IC_{X}) (intersection or Goresky-MacPherson homology) satisfy Poincaré duality. If X is a smooth algebraic variety (such as M^{G}(r, n)) or a quotient of a smooth variety by a finite group action (such as Sym^{n}(S)) then up to shift IC_{X} is just the constant sheaf ℂ (cf. [3] of [14]).

We will need a simplified version of Borho-MacPherson formula for the direct image of the intersection homology complex under a projective semismall morphism. This formula is a direct application of the Decomposition Theorem due to Beilinson-Berstein-Deligne-Gabber (cf. [3]).

Proposition 3.3. Suppose that a projective morphism Z → Y of algebraic varieties is strictly semi-small with respect to some stratification Y = ⋃ Y_{μ}. Suppose further that for any point y ∈ Y the fiber π^{−1}(y) is irreducible. Then

π_{∗}(IC_{Z}) = ⨁_{μ} IC_{\overline{Y}_{μ}}.

Corollary 3.4. One has the following direct sum decomposition in the derived category of complexes of sheaves:

(a_{r})_{∗}IC_{M^{G}(r,n)} = ⨁_{s∈\{0,...n\}, μ∈P (s)} IC_{\overline{M}_{s,μ}^{U}(r,n)}

where P (s) is the set of all partitions of s and IC(\overline{M}_{s,μ}^{U}) denotes the IC-complex on the closure of the stratum M_{s,μ}^{U} ⊂ M^{U}(r, n).

We intend to use the formula above by taking the cohomology of both sides. The IC-complexes supported on the closures of the smaller strata can be understood with the help of the following

Proposition 3.5. Let (s, μ) and m_{i} be as above and denote by Sym^{μ}(S) the direct product Sym^{m_{1}}(S) × … × Sym^{m_{n}}(S). Then there exists a finite birational morphism respecting the induced stratifications:

π_{n,s,μ} : M^{U}(r, n − s) × Sym^{μ}(S) → \overline{M}_{s,μ}^{U}(r, n)

Proof. Left as an exercise to the reader. q.e.d.

Now [14, Lemma 1] implies that in this situation one can deduce a

Corollary 3.6. In the notation of the previous proposition, one has the following equality in the derived category of sheaves:

(π_{n,s,μ})_{∗}(IC_{M^{U} (r,n−s)×Sym^{μ}(S)}) = IC_{\overline{M}_{s,μ}^{U}(r,n)}.

Recall that intersection homology complex of a space X is defined in such a way that it has non-trivial (global) hypercohomology in the range between (−k) and k where 2k is the real dimension of X.

Definition. The shifted intersection homology Poincaré polynomial P_{t}(X) is defined by the formula

P_{t}(X) := ∑_{− dim_{ℂ} X}^{dim_{ℂ} X}(dim_{ℂ} H^{i}(X, IC_{X})) · t^{i}

Note that for a smooth X this is just the usual Poincaré polynomial multiplied by t^{− dim_{ℂ} X}.

Theorem 3.7. One has the following identity between Poincaré polynomials of M^{G} and M^{U}:

\frac{∑_{n=0}^{∞}q^{n}P_{t}(M^{G}(r, n))}{∑_{n=0}^{∞} q^{n}P_{t}(M^{U}(r, n))} = ∏_{l=1}^{∞} \frac{(1 + t^{−1}q^{l})^{b_{1}(S)}(1 + t^{1}q^{l})^{b_{3}(S)}}{(1 − t^{−2}q^{l})^{b_{0}(S)}(1 − q^{l})^{b_{2}(S)}(1 − t^{2}q^{l})^{b_{4}(S)}} .

In short, the series for M^{G} is obtained from the series for M^{U} when multiplied by the Göttsche’s formula.

Proof. Following Nakajima’s notations, denote by a_{m}(t) the (shifted) Poincaré polynomial P_{t}(Sym^{m}(S)). Then a formula due to MacDonald [23] says that

∑_{m=0}^{∞} q^{m}a_{m}(t) = \frac{(1 + t^{−1}q)^{b_{1}(S)}(1 + tq)^{b_{3}(S)}}{(1 − t^{−2}q)^{b_{0}(X)}(1 − q)^{b_{2}(S)}(1 − t^{2}q)^{b_{4}(S)}} .

Taking the cohomology of both sides in the formula of Corollary 3.4 and using Corollary 3.6 one obtains:

∑_{k=0}^{∞} q^{k}P_{t}(M^{G}(r, k)) = ∑_{k=0}^{∞} ∑_{s=0}^{k} ∑_{μ∈P (s)}q^{k}P_{t}(M^{U}(r, k − s)) · P_{t}(Sym^{μ}S)

= ∑_{k=0}^{∞} ∑_{{0≤s≤k \atop μ∈P (s)}}q^{k−s}P_{t}(M^{U}(r, k − s)) · q^{s}P_{t}(Sym^{μ}S)

= ( ∑_{n=0}^{∞} q^{n}P_{t}(M^{U}(r, n))) · ( ∑_{{s≥0 \atop μ∈P (s)}} q^{s}P_{t}(Sym^{μ}S)).

Represent all partitions μ in the formula above as (1^{m_{1}}, 2^{m_{2}}, …, s^{m_{s}}). Then

∑_{{s≥0 \atop μ∈P (s)}} q^{s}P_{t}(Sym^{μ}S) = ∑_{{s≥0 \atop μ∈P (s)}} a_{m_{1}}(t)(q)^{m_{1}}a_{m_{2}}(t)(q^{2})^{m_{2}} … a_{m_{s}}(t)(q^{s})^{m_{s}}

= ∏_{l=1}^{∞} ( ∑_{m=0}^{∞} a_{m}(t)(q^{l})^{m})

= ∏_{l=1}^{∞} \frac{(1 + t^{−1}q^{l})^{b_{1}(S)}(1 + t^{1}q^{l})^{b_{3}(S)}}{(1 − t^{−2}q^{l})^{b_{0}(S)}(1 − q^{l})^{b_{2}(S)}(1 − t^{2}q^{l})^{b_{4}(S)}} .

q.e.d.

Remarks.

(1) A very similar formula was discovered by Göttsche in [13]. He computes the ratio of generating functions for M^{G}(r, n) and N(r, n) and obtains a similar product (involving more factors).

(2) In [29] Li and Qin derive a formula relating cohomology of Gieseker spaces for S and its blowup S̃. Applying Theorem 3.7 we get a relation between the intersection homology of Uhlenbeck compactifications for S and S̃. This formula is different form the relation between usual homology of M^{U}(r, n) for S, S̃ (also found by Li and Qin), since Uhlenbeck compactifications are quite singular hence their intersection homology is different from the ordinary (co)homology.

(3) Since S is compact, one of course has b_{0} = b_{4} = 1 and b_{1} = b_{3}. However, Göttsche-Soergel’s approach is valid for Hilbert schemes of any quasi-projective surface. It is interesting to know if the computaion above can be extended to the quasi-projective case. For example, one can consider S = ℂ^{2} viewed as an open subset of ℂℙ^{2} and work with the moduli spaces of sheaves on ℂℙ^{2} equipped with a trivialization on the infinite line l = ℂℙ^{2} \ ℂ^{2}. In this particular example the argument used in the proof of the theorem carries over without any changes, but the general case remains open.

Following [30, 1.13] one can define pure Hodge structure on the intersection homology of any complex algebraic variety X. Then, similarly to the usual shifted Poincaré polynomial, one defines a (shifted) Hodge polynomial P_{x,y}(X) using the pure Hodge structure on IH^{∗}(X). When X is smooth this coincides with the usual Hodge polynomial multiplied by (xy)^{− dim_{ℂ} X/2}. However, in general it will not be a shift of the virtual Hodge polynomial of X.

Using Saito’s theory of mixed Hodge modules (cf. [30]) and the formula for Hodge polynomials of symmetric powers (cf. [6]) one can repeat the arguments above to prove the following

Proposition 3.8.

(a) There exists an isomorpshism of pure Hodge structures:

IH^{∗}(M^{G}(r, n)) ≃ ⨁_{{s≥0 \atop μ∈P (s)}} IH^{∗}(M^{U}(r, n − s)) ⊗ IH^{∗}(Sym^{μ}(S));

(b) One has the following identity between the intersection homology Hodge polynomials of M^{G}(r, n) and M^{U}(r, n):

\frac{∑_{n≥0}P_{x,y}(M^{G}(r, n))q^{n}}{∑_{n≥0}P_{x,y}(M^{U}(r, n))q^{n}}

= ∏_{l≥0} \frac{(1+x^{−1}q^{l})^{h^{1,0}}(1+y^{−1}q^{l})^{h^{0,1}}(1+xq^{l})^{h^{2,1}}(1+yq^{l})^{h^{1,2}}}{(1−(xy)^{−1}q^{l})^{h^{0,0}}(1−xy^{−1}q^{l})^{h^{2,0}}(1−q^{l})^{h^{1,1}}(1−x^{−1}yq^{l})^{h^{0,2}}(1−xyq^{l})^{h^{2,2}}}

where h^{r,s} are the (usual) Hodge numbers of S.

4. Correspondences

In this section we give a very natural generalization of correspondences used by Nakajima and Grojnowski and prove that their action on the cohomology satisfies the correct commutation relations. Of course, as one might guess from the formula of Theorem 3.7, the corresponding module over the oscillator algebra is not irreducible any more. In fact, the space of vacuum vectors can be naturally identified with intersection homology of Uhlenbeck compactifications.

We now briefly recall the definition of the convolution product, cf. [7]. Let M_{1}, M_{2} be two smooth projective varieties and Z be a subvariety of M_{1} × M_{2}, such that restrictions of the two projections p_{1} : Z → M_{1}, p_{2} : Z → M_{2} are proper. Such Z defines the map P_{Z} : H^{∗}(M_{1}) → H^{∗}(M_{2}) by the formula

P_{Z}(α) = (p_{2})_{∗}(p_{1}^{∗}α),

More generally, let β ∈ H^{∗}(Z) be a cohomology class (in an appropriate cohomology theory if Z is singular). Then we can consider the convolution operator P_{Z}^{β} : H^{∗}(M_{1}) → H^{∗}(M_{2}) defined by

P_{Z}^{β}(α) = (p_{2})_{∗}(β ∪ p_{1}^{∗}α).

Of course, we can switch the roles of M_{1} and M_{2} and define the map in the other direction.

If M_{3} is a third smooth projective variety, Z′ ⊂ M_{2} × M_{3} is a subvariety which projects properly to M_{2} and M_{3}, and β′ ∈ H^{∗}(Z′) then the composition P_{Z′}^{β′} ∘ P_{Z}^{β} : H^{∗}(M_{1}) → H^{∗}(M_{3}) can be described as follows. Consider the subvariety Z̃ ⊂ M_{1} × M_{3} given by

Z̃ = p_{13}(p_{12}^{−1}(Z) ∩ p_{13}^{−1}(Z′)),

where p_{12} is the projections M_{1} × M_{2} × M_{3} → M_{1} × M_{2}, and similarly for p_{13} and p_{23}. We define the cohomology class β̃ ∈ H^{∗}(Z̃) as (p_{13})_{∗}(p_{12}^{∗}(β)∪p_{23}^{∗}(β′)). Then, (cf. [7, Chapter 3]), we have the equality

P_{Z′}^{β′} ∘ P_{Z}^{β} = P β̃ ; Z̃ : H^{∗}(M_{1}) → H^{∗}(M_{3}).

We now describe the subvarieties in the products of the moduli spaces, to which the above convolution construction will be applied.

Definition. For i > 0, define a subvariety

P_{−i} ⊂ ∐_{n} M^{G}(r, n) × M(r, n + i)

as a set of all pairs (F_{1}, F_{2}) such that:

(1) F_{1}^{∗∗} ≃ F_{2}^{∗∗};

(2) F_{2} ⊂ F_{1} as subsheaves of the common double dual;

(3) Supp(F_{1}/F_{2}) = {x} for some x ∈ X. Similarly, we define P_{i} ⊂ ∐_{n} M^{G}(r, n + i) × M(r, n) for i > 0 by ex-

changing the roles of F_{1} and F_{2}.

Remark. Since F_{1} and F_{2} are stable, so is F := F_{1}^{∗∗} ≃ F_{2}^{∗∗}. This implies Hom(F, F) = ℂ, and the embeddings F_{1} ↪ F, F_{2} ↪ F are defined canonically up to multiplication by a scalar. This ensures that the second condition is well-defined.

One has a natural morphism Π : P_{i} → S given by

Π(F_{1}, F_{2}) = x for (F_{1}, F_{2}) ∈ P_{i},

where x is the unique element of Supp(F_{1}/F_{2}). One can give a rigorous definition of Π using universal sheaves on moduli spaces and [17, Example 4.3.6] but we will not do it here.

Denote the dimension of M^{G}(r, n) by 2rn + a where a is a number depending on S, r, L and H but not on n, cf. (2). Then the dimension of P_{−i} ∩ M^{G}(n − i) × M^{G}(n) is equal to 2rn + a − ri + 1 (this follows from Nakajima’s argument [27, 8.3] and Theorem 2.2).

Let P_{±i}^{α} : ⨁_{n} H^{∗}(M^{G}(r, n)) → ⨁_{n} H^{∗}(M^{G}(r, n ∓ i)), for α ∈ H^{∗}(S)

be the convolution operators induced by the classes Π^{∗}(α) ∈ H^{∗}(P_{±i}). Computing the dimensions we can see that P_{−i}^{α} increases the cohomological degree by 2(ri − 1) + deg α.

The next theorem asserts that ⨁_{n} H^{∗}(M^{G}(r, n)) becomes a repre-

sentation space of the Heisenberg/Clifford algebra generated by P_{i}^{α} and P_{−i}^{β}/(−1)^{ri−1}r. It is a direct generalization of [27, Theorem 8.13].

Theorem 4.1. The following relations hold:

[P_{i}^{α}, P_{j}^{β}] = (−1)^{ri−1}ri δ_{i+j,0}⟨α, β⟩ Id if (−1)^{degα·deg β} = 1,
{P_{i}^{α}, P_{j}^{β}} = (−1)^{ri−1}ri δ_{i+j,0}⟨α, β⟩ Id otherwise,

where ⟨α, β⟩ is the intersection form on S, [A, B] = AB − BA and {A, B} = AB + BA.

Exactly as in [27, Corollary 8.16] we deduce:

Corollary 4.2. The cohomology groups ⊕_{n=0}^{∞}H^{∗}(M^{G}(r, n)) form a tensor product of an irreducible highest weight representation of the oscillator algebra, with a trivial representation on the intersection homology groups of Uhlenbeck compactifications.

Example. Let r = 2, i = 1, S = ℙ^{2} and L ≃ H ≃ O(1). Choose α to be the fundamental class and β the class of a point x ∈ ℙ^{2}. Then the formula above predicts that [P_{1}^{α}, P_{−1}^{β}] = (−2)Id. This can be seen directly.

It is known that M^{G}(2, 1) is a single point corresponding to the twist T_{ℙ^{2}}(−1) of the tangent bundle (cf. [28, 3.2]). The Uhlenbeck compactification M^{U}(2, 2) can be described as follows. Let V be the 6dimensional space of symmetric 3 × 3 matrices. Then M^{U}(2, 2) can be identified with a hypersurface H in P (V) correspoding to all rank ≤ 2 symmetric matrices (cf. [28, 4.3]). Its singular locus P ≃ ℙ^{2} corresponds to symmetric matrices of rank one. It can be deduced from (cf. [28, 4.3]) that M^{G}(2, 2) is isomorphic to the blowup of P.

Let D ⊂ M^{G}(2, 2) be the exceptional divisor. Each one-dimensional fiber of D → P corresponds to the set of sheaves F which fit a short exact sequence

0 → F → T_{ℙ^{2}}(−1) → ℂ_{x} → 0,

where x ∈ ℙ^{2} ≃ P. Let l be such a fiber and [pt] the generator of H^{0}(M^{G}(2, 1)). Since P_{1}^{α}([pt]) = 0 and P_{−1}^{β}([pt]) = [l] we expect that P_{1}^{α}([l]) = −2[pt]. It follows from definitions that this is equivalent to [D] · [l] = −2. The last equality is true since H has an A_{2}-singularity at any point of P.

Now we begin to prove Theorem 4.1. Most of Nakajima’s proof, cf. [27, Chapter 8], can be repeated word-by-word with almost no changes. In particular, we have the following result:

Theorem 4.3. For any v ∈ H^{∗}(M^{G}(r, n)), the following relations hold:

[P_{i}^{α}, P_{j}^{β}](v) = c_{i,n} δ_{i+j,0}⟨α, β⟩ v if (−1)^{deg α·deg β} = 1,
{P_{i}^{α}, P_{j}^{β}}(v) = c_{i,n} δ_{i+j,0}⟨α, β⟩ v otherwise,

where ⟨α, β⟩ is the intersection form on H^{∗}(S) and c_{i,n} is some constant depending apriori on i and n but not on the classes α and β.

The proof is a mere repetition of the argument presented in [27, 8.4] (where all the dimension statements are proved using Theorem 2.2).

Lemma 4.4. For each i, the constants c_{i,n} are independent of n.

Proof. Let k ≠ ±i. By Theorem 4.3 we have [P_{−k}^{γ}, [P_{i}^{α}, P_{−i}^{β}]] = 0. Take v ∈ H^{∗}(M^{G}(r, n)). Applying the double commutator to v, one gets

⟨α, β⟩(c_{i,n} − c_{i,n+k})P_{−k}^{γ}v = 0.

One can choose v, α, β, γ in such a way that ⟨α, β⟩ ≠ 0 and P_{γ}[−k](v) ≠ 0. Therefore, one has c_{i,n} = c_{i,n+k} if k ≠ ±i. If i ≠ ±1 we can take k = 1 and obtain c_{i,n} = c_{i,n+1}. If i = ±1 we take k = 2, 3 and get c_{1,n} = c_{1,n+2} = c_{1,n+3} which also implies that c_{1,n} is independent of n. q.e.d.

In view of this lemma we will be writing c_{i} instead of c_{i,n}.

Recall the general setup of Nakajima’s Chapter 9 (or rather its adaptation to our case). We choose and fix two smooth transversal curves C and C′ in one very ample linear system on S. Take such an s that M^{G}(r, s) is non-empty and fix a locally free sheaf E correspoding to a point in M^{G}(r, s).

We will compute numbers c_{i} using the formalism of generating functions and a connection with symmetric functions discovered by Nakajima. The actual idea of using embedded curves is originally due to Grojnowski.

To that end, consider the subvariety L^{k,s} ⊂ M^{G}(r, n) formed by all sheaves F such that:

(a) F^{∗∗} ≃ E;

(b) The quotient A_{F} := E/F is an Artin sheaf of length k = n − s supported at finitely many points of C.

Definition. For each partition μ = (μ_{1} ≥ … ≥ μ_{l} > 0) of k let (L^{μ,s})^{∘} be the set of all sheaves F such that:

(a) F^{∗∗} ≃ E;

(b) The quotient A_{F} = E/F is supported at pairwise distinct points x_{j} ∈ C, 1 ≤ j ≤ l with multiplicities μ_{j}.

Denote also by L^{μ,s} the closure of (L^{μ,s})^{∘}.

Definition. Similarly, we define L̂^{μ,s} to be the closure of the subset of all sheaves F satisfying:

(a) F ⊂ F′ for some F′ ∈ M^{G}(r, s);

(b) the quotient F′/F is supported at pairwise distinct points x_{j} ∈ C, 1 ≤ j ≤ l with multiplicities μ_{j}.

Proposition 4.5. The closed subvarieties L^{μ,s} are the irreducible components of L^{k,s}. Each component L^{μ,s} is of dimension rk. Similarly, L̂^{μ,s} is the set of all irreducible components of L̂^{k,s} and all L̂^{μ,s} are of the same dimension.

Proof. If we fix all the points x_{j} is the definition of L^{μ,s} then by Theorem 2.2 the variety of all sheaves with the required condition has dimension ∑_{i=1}^{l}(rμ_{i} − 1) = rk − l. Since the points x_{j}, j = 1, …, l are allowed to move in an l-dimensional family, we obtain dim L^{μ,s} = rk. By the irreducibility part of Theorem 2.2 we conclude that each L^{μ,s} is irreducible.

It follows from the definition of L^{k,s} that it is a union of the closed subsets L^{μ,s}. Hence, L^{μ,s} are the irreducible components of L^{k,s}. q.e.d.

Note that P_{−i}^{C} maps classes of (cohomological) degree b on M^{G}(r, n) to classes of degree ri + b on M^{G}(r, n + i). Thus, the subspace ⨁ H^{rn}(M^{G}(r, n)) is preserved by P_{−i}^{C}. Moreover, the next theorem shows that in fact even the smaller subspace generated by the classes of L^{μ,s}, is invariant under P_{−i}^{C} (i > 0).

Theorem 4.6 (cf. [27, Thm. 9.14]). For i > 0, we have

P_{−i}^{C}[L^{μ,s}] = ∑_{λ} a_{λμ}[L^{λ,s}],

where the summation is over partitions λ of |μ| + i which are obtained as follows:

(a) add i to a term in μ, say μ_{j} (possibly 0), and then

(b) arrange it in descending order.

The coefficient a_{λμ} is #{l | λ_{l} = μ_{j} + i}.

A similar statement is true for the classes of L̂^{μ,s}.

This theorem will be proved in Section 5.

Let [vac] be the cohomology class of the point E ∈ M^{G}(r, s) and [V ac] be the fundamental class of M^{G}(r, s). Theorem 4.6 allows us to establish a connection with the theory of symmetric functions as follows. Let L (resp. L̂) be the subspace in ⨁_{n} H^{∗}(M^{G}(r, n)) generated by the classes of L^{μ,s} and [vac] (resp. L̂^{μ,s} and [V ac]). We define a ℂ-linear isomorphism from L (resp. L̂) onto the space Λ of symmetric functions in infinitely many variables (cf. [27, 9.1], [24]). This isomorphism sends [vac] (resp. [V ac]) to 1 ∈ Λ and [L^{μ,s}] (resp. [L̂^{μ,s}]) to the orbit sum function m_{λ} (cf. loc. cit.).

Theorem 4.6 means that the operator P_{−i}^{C} corresponds under the isomorphism above to multiplication by the i-th power sum (or Newton function) p_{i} ∈ Λ (loc. cit.). This provides us with several identities between cohomology classes coming from classical identities between symmetric functions.

Note that for μ = (1^{k}) the corresponding variety L^{μ,s} is isomorphic to the global Quot scheme Quot_{C}^{k}(E) on the curve C. This scheme parametrizes quotient sheaves E|_{C} → A on C, where A is of length k. Every such quotient on C defines a sheaf on S, namely the kernel of the composition E → E|_{C} → A. The class of Quot_{C}^{k}(E) corresponds to the k-th elementary symmetric function e_{k} = m_{(1^{k})}.

Similarly, we will slightly abuse notation by writing Quot_{C}^{k}(s) instead of L̂^{(1^{k}),s}. This variety is a birational image of the family of schemes Quot_{C}^{k}(F′) parametrized by points F′ of M^{G}(r, s).

Repeating the arguments of Nakajima, (see [27, formula 9.16]) we obtain the formulas

(3)  ∑_{n=0}^{∞} z^{n}[Quot_{C}^{n}(E)] = exp( ∑_{i=1}^{∞} \frac{z^{i}P_{i}^{C}}{(−1)^{i−1}i}) · [vac]

(4)  ∑_{n=0}^{∞} z^{n}[Quot_{C}^{n}(s)] = exp( ∑_{i=1}^{∞} \frac{z^{i}P_{i}^{C}}{(−1)^{i−1}i}) · [V ac].

These formulas arise from the classical identity between Newton symmetric functions and elementary symmetric functions (cf. [24]). They are the first ingredient in our computation of c_{i}. The second ingredient is provided by

Theorem 4.7 (cf. [27, Exercise 9.23]). The following relation holds:

∑_{n=0}^{∞} z^{2n}⟨Quot_{C}^{n}(E), Quot_{C′}^{n}(s)⟩ = (1 − (−1)^{r}z^{2})^{r⟨C,C′⟩}.

The proof of this theorem will be given in Section 6 for the case when the intersection C ⋂ C′ is transversal (which is sufficient for our purposes). Then the general case can be deduced from Theorem 4.1 via the argument of the next section.

End of the proof of Theorem 4.1. Following Nakajima, we introduce the following notations:

C_{−}(z) = ∑_{i=1}^{∞} \frac{P_{−i}^{C}z^{i}}{(−1)^{i−1}i}; C_{+}(z) = ∑_{i=1}^{∞} \frac{P_{i}^{C}z^{i}}{(−1)^{i−1}i}.

Note that C_{−}(z) is adjoint to C_{+}(z) with respect to the intersection form on the direct sum ⨁_{n} H^{∗}(M^{G}(r, n)) (the cohomology groups for

different n are orthogonal to each other). We extend this form to power series in z by z-linearity. By Theorem 4.7 and (3), (4) one has:

(1 − (−1)^{r}z^{2})^{r⟨C,C′⟩} = ∑_{n=0}^{∞} z^{2n}⟨Quot_{C}^{n}(E), Quot_{C′}^{n}(s)⟩

= ⟨∑_{n=0}^{∞} z^{n}Quot_{C}^{n}(E), ∑_{n=0}^{∞} z^{n}Quot_{C′}^{n}(s)⟩
= ⟨exp(C_{−}(z)) · [vac], exp(C′_{−}(z)) · [V ac]⟩
= ⟨exp(C′_{+}(z)) exp(C_{−}(z)) · [vac], [V ac]⟩

where the last equation follows from adjointness.

We will use the following identity which holds for any pair of operators A and B:

exp(−A) B exp(A) = exp(−ad A)(B) = 1−(ad A)(B)+\frac{(ad A)^{2}}{2!} (B)−…

Putting A = C_{−}(z), B = exp(C′_{+}(z)) we obtain

⟨exp(C′_{+}(z)) exp(C_{−}(z)) · [vac], [V ac]⟩ =

= ⟨exp(C_{−}(z))[exp(−ad C_{−}(z))(exp(C′_{+}(z)))] · [vac], [V ac]⟩.

An explicit computation shows that

[C_{−}(z), exp(C′_{+}(z))] = −( ∑_{n=1}^{∞} \frac{c_{n}}{n^{2}} ⟨C, C′⟩z^{2n}) exp(C′_{+}(z)).

Denote the expression ∑_{n=1}^{∞} \frac{c_{n}}{n^{2}} ⟨C, C′⟩z^{2n} by Φ(z). Then by the previous

formula:

exp(−ad C_{−}(z))(exp(C′_{+}(z))) = exp(Φ(z))exp(C′_{+}(z)).

Collecting the results of computations, we obtain:

(1 − (−1)^{r}z^{2})^{r⟨C,C′⟩} = exp(Φ(z))⟨exp(C_{−}(z)) exp(C′_{+}(z)) · [vac], [V ac]⟩
= exp(Φ(z))⟨exp(C_{−}(z)) · [vac], [V ac]⟩
= exp(Φ(z)).

The last equality holds since all P_{−i}^{C} involved in the definition of C_{+}(z), map [vac] to the orthogonal complement of [V ac]. By definition of Φ(z) we have

∑_{n=1}^{∞} \frac{c_{n}}{n^{2}} ⟨C, C′⟩z^{2n} = r⟨C, C′⟩log(1 − (−1)^{r}z^{2}).

Hence c_{n} = (−1)^{rn−1}rn which completes the proof of Theorem 4.1. q.e.d.

5. A transversality result

In the case r = 1, Theorem 4.6 has a simple proof (cf. [27]) using local coordinates on S. While this result is still true in higher ranks the proof of it requires a more detailed analysis of the geometry of the moduli space to be provided below.

First consider the situation at the set-theoretic level. It follows for the definitions that if (F_{1}, F_{2}) ∈ P_{−i}^{C} then F_{1}^{∗∗} ≃ F_{2}^{∗∗} and Supp(F_{1}/F_{2}) = {x} ∈ C. In particular, if F_{1} ∈ L^{μ,s} then F_{2} necessarily belongs to one of the L^{λ,s} where the partition λ is as described in the theorem. Hence P_{−i}^{C}[L^{μ,s}] is a linear combination of [L^{λ,s}] with integral coefficients.

Our goal is to show that for generic F_{2} ∈ L^{λ,s} the intersection

p_{1}^{−1}(L^{μ,s}) ⋂ P_{−i}^{C} ⋂ p_{2}^{−1}(F_{2}) ⊂ M^{G}(r, s + |μ|) × M^{G}(r, s + |λ|)

is finite and consists of exactly a_{λμ} points.

Recall that by definition all sheaves F representing the points of L^{λ,s} satisfy F^{∗∗} ≃ E where E is a fixed vector bundle. In particular, this holds for F_{1} ∈ L^{μ,s} and F_{2} ∈ L^{λ,s}. Consider the quotient A_{F_{2}} = E/F_{2}. If λ = (λ_{1} ≥ ⋯ ≥ λ_{p} > 0) then A_{F_{2}} is supported at some points x_{j} ∈ C, 1 ≤ j ≤ p with multiplicities λ_{j}. Therefore one can write A_{F_{2}} as a direct sum ⨁(A_{F_{2}})_{x_{j}}.

We claim that for generic F_{2} there exist local coordinates (ζ_{j}, ξ_{j}) at x_{j} such that (A_{F_{2}})_{x_{j}} ≃ ℂ[ζ_{j}]/(ζ_{j})^{λ_{j}}.

In fact, choose a trivialization E ≃ O^{⊕r} in the neighbourhood of x_{j} and fix some system of coordinates (ζ′_{j}, ξ′_{j}) centered at x_{j}. Consider such quotients O^{⊕r} → A of length λ_{j} at x_{j} that the first component O → A is surjective and its kernel is generated by m_{x_{j}}^{λ_{j}} and some element ξ_{j} = ξ′_{j} + ∑_{k=1}^{λ_{j}−1} a_{k}(ζ′_{j})^{k}, where a_{1}, …, a_{λ_{j}−1} are arbitrary complex numbers. The other components O^{⊕(r−1)} → A can be chosen arbitrarily. If we take ζ_{j} = ζ′_{j} then A has the required form in the local coordinates (ζ_{j}, ξ_{j}). Thus we obtain a (rλ_{j}−1)-dimensional family of non-isomorphic quotients. Here λ_{j} − 1 parameters come from the choice of ξ_{j} and the other (r − 1)λ_{j} parameters come from the choice of the map O^{⊕(r−1)} → A. By Theorem 2.2, this family forms a dense subset in Quot(r, λ_{j}) and this implies our assertion.

Hence we can assume that E/F_{2} satisfies the conditions described above. Then a choice of F_{1} such that (F_{1}, F_{2}) ∈ P_{−i}^{C} amounts to choosing a (λ_{j} − i)-dimensional quotient (A_{F_{1}})_{x_{j}} of (A_{F_{2}})_{x_{j}} ≃ ℂ[ζ_{j}]/(ζ_{j})^{λ_{j}}. If x_{j} is fixed, this can be done in a unique way. Therefore the number of points in p_{1}^{−1}(L^{μ,s}) ⋂ P_{−i}^{C} over a generic F_{2} is equal to the number of ways to subtract i from one of the parts of partition λ, and obtain partition μ. This number is exactly a_{λμ}.

To finish the proof of Theorem 4.6 we need to show that p^{−1}(L^{μ,s}) intersects P_{−i}^{C} transversely. To that end, we prove the following lemma

Lemma 5.1. Let (F_{1}, F_{2}) ∈ P_{−i}^{C} ⋂ M^{G}(r, s + |μ|) × M^{G}(r, s + |λ|) be a smooth point of P_{−i}^{C} and assume that F_{1} ∈ L^{μ,s} (resp. F_{2} ∈ L^{λ,s}) is generic in the sense described above. Then the intersection

W = T_{(F_{1},F_{2})} p_{1}^{−1}(L^{μ,s}) ⋂ T_{(F_{1},F_{2})} P_{−i}^{C}

of tangent spaces to p_{1}^{−1}(L^{μ,s}) and P_{−i}^{C} projects isomorphically under p_{2} onto the tangent space T_{F_{2}}(L^{λ,s}).

Proof. It suffices to prove that if (v, w) ∈ W then w ∈ T_{F_{2}}(L^{λ,s}) and v is uniquely defined by w. Then the dimension count shows that in fact the map dp_{2} : W → T_{F_{2}}(L^{λ,s}) is an isomorphism. Our proof consists of several steps.

Step 1. First we compute the tangent spaces T_{F_{1}}(L^{μ,s}) and T_{F_{2}}(L^{λ,s}).

To that end, recall (cf. [1]) that the tangent space T_{F_{1}}(M^{G}(r, s+|μ|)) is isomorphic to the kernel of the natural trace map tr^{1} : Ext_{S}^{1}(F_{1}, F_{1}) → H^{1}(S, O). Infinitesimal deformations of F_{1} with fixed E = F_{1}^{∗∗} correspond to the subspace Hom_{S}(F_{1}, E/F_{1}) ⊂ Ext_{S}^{1}(F_{1}, F_{1}). The embedding of this subspace is just the boundary map which comes from applying Hom(F_{1}, · ) to a short exact sequence 0 → F_{1} → E → E/F_{1} → 0.

Since E/F_{1} = A_{F_{1}} = ⨁_{x_{j}∈Supp(A_{F_{1}})}(A_{F_{1}})_{x_{j}}, one has a direct sum

decomposition

Hom_{S}(F_{1}, A_{F_{1}}) = ⨁_{x_{j}∈Supp(A_{F_{1}})} Hom_{S}(F_{1}, (A_{F_{1}})_{x_{j}}).

Recall that μ_{j} = mult_{x_{j}}(A_{F_{1}}). Suppose that F_{1} is generic in the sense that it has a local description given in the previous section. Then locally in the neighbourhood of x_{j} ∈ S we have embeddeings E(−μ_{j}C) ⊂ F_{1} ⊂ E. Therefore the space Hom_{S}(F_{1}, (A_{F_{1}})_{x_{j}}) contains a subspace Hom_{S}(F_{1}/E(−μ_{j}C), (A_{F_{1}})_{x_{j}}). Now it is easy to see that the tangent space T_{F_{1}}(L^{μ,s}) corresponds precisely to the subspace

⨁_{x_{j}∈Supp(A_{F_{1}})} Hom_{S}(F_{1}/E(−μ_{j}C), (A_{F_{1}})_{x_{j}}) ⊂ Hom_{S}(F_{1}, A_{F_{1}})
⊂ Ext_{S}^{1}(F_{1}, F_{1}).

Similar computation applies to F_{2} ∈ L^{λ,s}.

Step 2. Now we assume that

(F_{1}, F_{2}) ∈ P_{−i}^{C} ⊂ M^{G}(r, s + |μ|) × M^{G}(r, s + |λ|)

is a generic point. We want to describe the tangent space to P_{−i}^{C} at the point (F_{1}, F_{2}), as a subspace of Ext_{S}^{1}(F_{1}, F_{1}) ⊕ Ext_{S}^{1}(F_{2}, F_{2}).

Recall that the vectors in Ext_{S}^{1}(F_{1}, F_{1}) correspond to extensions 0 → F_{1} → G_{1} → F_{1} → 0 which may be viewed as deformations over the parameter space ℂ[ϵ]/ϵ^{2} (and similarly for F_{2}).

The conditions Supp (F_{1}/F_{2}) = x ∈ C and length(F_{1}/F_{2}) = i imply that F_{1}(−iC) ⊂ F_{2}. Moreover, the following diagram

\begin{matrix}F_{1}(−iC) & ⟶ & F_{2} \\ ∥ & & ↓ \\ F_{1}(−iC) & ⟶^{a} & F_{1} \end{matrix} (5)

commutes. Here the map a is the multiplication by i-th power of the local equation for C.

The diagram (5) should be preserved under infinitesimal deformation of the point (F_{1}, F_{2}) within P_{C}[−i]. This means that:

(a) There exists a commutative diagram:

0 ⟶ F_{2} ⟶ G_{2} ⟶ F_{2} ⟶ 0
↓  ↓  ↓
0 ⟶ F_{1} ⟶ G_{1} ⟶ F_{1} ⟶ 0.

It is a standard fact of homological algebra that this is equivalent to requiring that the images of v and w under the natural maps

Ext_{S}^{1}(F_{1}, F_{1}) → Ext_{S}^{1}(F_{2}, F_{1}) and Ext_{S}^{1}(F_{2}, F_{2}) → Ext_{s}^{1}(F_{2}, F_{1})

coincide.

(b) There exists a similar diagram of extensions corresponding to the embedding F_{1}(−iC) ⊂ F_{2}. This is can be expressed in terms of Ext groups in a similar way.

(c) The diagram of middile terms

G_{1}(−iC) ⟶ G_{2}
∥  ↓
G_{1}(−iC) ⟶^{a} G_{1}

commutes. This condition can be expressed as vanishing of some homomorphism from the quotient copy of F_{1}(−iC) to the subsheaf copy of F_{1} (i.e., the embedding of sheaves a should not be deformed). We will not write this down explicitly as we need this condition only in a special case (see below).

Step 3. We will prove that if (v, w) ∈ W (where W is the tangent space to the intersection p_{1}^{−1}(L^{μ,s}) ∩ P_{−i}^{C}), then w ∈ T_{F_{2}}(L^{λ,s}), and that w = 0 implies v = 0. The condition w ∈ T_{F_{2}}(L^{λ,s}) will follow from (a) above while the implication (w = 0) ⇒ (v = 0) is a consequence of (c).

To use (a), consider the diagram:

\begin{aligned}0 & ⟶ & F_{1} & ⟶ & E & ⟶ & A_{F_{1}} & ⟶ & 0 \\ & & ↑ & & ↑ & & ↑ \\ 0 & ⟶ & F_{2} & ⟶ & E & ⟶ & A_{F_{2}} & ⟶ & 0 \end{aligned} (6)

and the induced commutative digram of Ext-groups:

Hom_{S}(F_{1}, A_{F_{1}}) ⟶ Ext_{S}^{1}(F_{1}, F_{1}) ⟶ Ext_{S}^{1}(F_{1}, E)
↓  ↓  ↓
Hom_{S}(F_{2}, A_{F_{1}}) ⟶ Ext_{S}^{1}(F_{2}, F_{1}) ⟶ Ext_{S}^{1}(F_{2}, E)
↑  ↑  ∥
Hom_{S}(F_{2}, A_{F_{2}}) ⟶ Ext_{S}^{1}(F_{2}, F_{2}) ⟶ Ext_{S}^{1}(F_{2}, E).

Here the lower two rows are obtained by applying Hom_{S}(F_{2}, · ) to (6) and the upper row comes from applying Hom_{S}(F_{2}, · ) to the upper row of (6). Note that by stability Hom_{S}(E, E) = Hom_{S}(F_{i}, F_{i}) = Hom_{S}(F_{i}, E) = ℂ and hence the first arrow in each row is injective. Simple diagram chasing shows that if

v ∈ ⨁_{x_{j}∈Supp(A_{F_{1}})} Hom_{S}(F_{1}/E(−μ_{j}C), (A_{F_{1}})_{x_{j}}) ⊂ Hom_{S}(F_{1}, A_{F_{1}})

then by the condition (b) we have

w ∈ ⨁_{x_{j}∈Supp(A_{F_{2}})} Hom_{S}(F_{2}/E(−λ_{j}C), (A_{F_{2}})_{x_{j}}) ⊂ Hom_{S}(F_{2}, A_{F_{2}}).

To prove injectivity, recall that for

v ∈ Hom_{S}(F_{1}, A_{F_{1}}) ⊂ Ext^{1}(F_{1}, F_{1})

the extension

0 → F_{1} → G_{1} → F_{1} → 0

can be recovered as a kernel of the map of extensions

0 ⟶ E ⟶ E ⊕ F_{1} ⟶ F_{1} ⟶ 0
b↓  ^{(b⊕v)}↓  ↓
0 ⟶ A_{F_{1}}  A_{F_{1}} ⟶ 0 ⟶ 0

where b is the cokernel of the natural embedding F_{1} ⊂ E. Similar statement is true for w and the corresponding extension 0 → F_{2} → G_{2} → F_{2} → 0. One shows that in our case the condition (c) translates into commutativity of the diagram

F_{1}(−iC) ⟶^{v} A_{F_{1}}(−iC)
↓  ↓
F_{2}  ⟶^{w} A_{F_{2}}.

Since both vertical arrows in this diagram are injective the condition w = 0 implies v = 0.

This ends the proof of the lemma and hence the proof of Theorem 4.6. q.e.d.

6. Computation of the intersection number

The main result of this section is a computation of the intersection number of the fundamental classes of cycles Quot_{C}^{n}(E) and Quot_{C′}^{n}(s) in M^{G}(r, n+s). In the case of Hilbert schemes this intersection is transversal and the intersection number can be found by a simple set-theoretic argument (recall that we assume that C and C′ are transversal). However, for higher ranks the set-theoretic intersection is not transversal any more and to compute the intersection index we have to apply the excess intersection formula (cf. [11]).

By definition Quot_{C}^{n}(E) ⊂ M^{G}(r, n + s) parametrizes all short exact sequences 0 → E_{1} → E|_{C} → A → 0 of sheaves on C, where A is a length n Artin sheaf. Any such sequence defines a sheaf F ∈ M^{G}(r, n + s) on S, namely the kernel of the composition E → E|_{C} → A. Note that Quot_{C}^{n}(E) is smooth since the tangent space to it at a point represented by 0 → E_{1} → E|_{C} → A → 0 is Hom_{O_{C}}(E_{1}, A) which has a constant dimension rn since E_{1} is a rank r vector bundle on C, and A is an Artin sheaf on C.

Similarly, Quot_{C′}^{n}(s) ⊂ M^{G}(r, n + s) has open subset (Quot_{C′}^{n}(s))^{∘} isomorphic to a fiber bundle over the subset (M^{G}(r, s))^{∘} of locally free sheaves, with fiber Quot_{ℂ′}^{n}(E′) over E′ ∈ (M^{G}(r, s))^{∘}. Hence (Quot_{C′}^{n}(s))^{∘} is smooth.

Note that Quot_{C′}^{n}(s)\(Quot_{C′}^{n}(s))^{∘} does not intersect Quot_{C}^{n}(E) since all points

F ∈ Quot_{C}^{n}(E) ⊂ M^{G}(r, n + s)

satisfy length(F^{∗∗}/F) = n and for sheaves in Quot_{C′}^{n}(s) \ (Quot_{C′}^{n}(s))^{∘} this length is at least (n + 1).

Assume for the sake of simplicity that S = ℙ^{2} and C, C′ are two distinct lines intersecting at a point x ∈ S. The general case follows from our proof by a simple combinatorial argument.

Let F ∈ Quot_{C′}^{n}(s) ⋂ Quot_{C}^{n}(E). Then F^{∗∗} = E, F is a kernel of E → E|_{C} → A and also a kernel of E → E|_{C′} → A′. Hence F contains E(−C) + E(−C′) = E ⊗ J_{x} as a subsheaf and A ≃ A′ ≃ (ℂ_{x})^{n}, where J_{x} is the ideal sheaf of x ∈ S and ℂ_{x} ≃ O_{S}/J_{x}. This means that every sheaf F in our intersection can be obtained as a kernel of E → E_{x} → A where A is an n-dimensional quotient of an r-dimensional vector space E_{x}. Hence Quot_{C′}^{n}(s) ⋂ Quot_{C}^{n}(E) is nothing but the Grassman variety Gr(E_{x}, n) of all n-dimensional quotients of E_{x}. In particular, n ≤ r.

Since Quot_{C′}^{n}(s) and Quot_{C}^{n}(E) are of complementary dimensions in the ambient M^{G}(r, n + s) we see that for r ≥ 2 their set-theoretic intersection has abnormally high dimension. However, it is true that Quot_{C′}^{n}(s) and Quot_{C}^{n}(E) are smooth along the points of Gr(E_{x}, n) and, moreover, the intersection is clean, i.e., for any F ∈ Gr(E_{x}, n)

T_{F} Gr(E_{x}, n) = T_{F} Quot_{C′}^{n}(s) ⋂ T_{F} Quot_{C}^{n}(E)

(this can be checked using methods of the previous section).

Therefore, we can apply excess intersection formula (cf. [11, Example 6.1.7]): let V be a rank (r −n)n vector bundle on Gr(E_{x}, n) arising form the following exact sequence:

0 → T_{Gr(r,n)} → T_{Quot_{C}^{n}(E)} ⊕ T_{Quot_{C′}^{n}(s)} → T_{M^{G}(r,n+s)} → V → 0. (7)

Then the intersection number [Quot_{C}^{n}(E)] · [Quot_{C′}^{n}(s)] in M^{G}(r, n + s) is equal to the top Chern class c_{(r−n)n}(V).

To compute the Chern class of the previous section we need to consider certain sheaves on M^{G}(r, n + s) × S.

Notation. For any pair of sheaves G_{1}, G_{2} on M^{G}(r, n + s) × S let Ext_{p_{1}}^{i}(G_{1}, G_{2}) be the relative Ext-sheaf on M^{G}(r, n + s) with respect to the first projection p_{1} : M^{G}(r, n + s) × S → M^{G}(r, n + s).

We will only deal with the cases when for all x ∈ M^{G}(r, n + s) the global Ext group Ext^{i}(p_{1}^{−1}(x); G_{1}, G_{2}) on the fiber p_{1}^{−1}(x) is of constant dimension. Hence by [19] Ext_{p_{1}}^{i}(G_{1}, G_{2}) is a vector bundle on M^{G}(r, n + s) which has the global Ext-group above as the fiber over point x. Similar remarks apply to any closed subspace of M^{G}(r, n + s) (abusing notation we will denote a sheaf on M^{G}(r, n+s) and its restriction to a closed subspace by the same letter).

Since gcd(r, c_{1}(H) · c_{1}(L)) = 1, there exists a universal sheaf G on M^{G}(r, n+s)×S (cf. [17]). It is known (cf. [22]) that the tangent bundle T_{M^{G}(r,n+s)} is isomorphic to the kernel of the surjective map of the vector bundles

Ext_{p_{1}}^{1}(G, G) → H^{1}(S, O_{S}) ⊗ O_{M^{G}(r,n+s)} → 0.

In particular, the tangent bundle to M^{G}(r, n + s) has the same Chern classes as Ext_{p_{1}}^{1}(G, G).

Lemma 6.1. Let Q be the universal quotient bundle on the Grassman variety Gr(E_{x}, n). The the full Chern class of Ext_{p_{1}}^{1}(G, G)|_{Gr(E_{x},n)} is equal to (c(Q)c(Q^{∗}))^{r}, where c(Q) (resp. c(Q^{∗})) is the full Chern class of Q (resp. its dual).

Proof. Note that G|_{Gr(E_{x},n)×S} can be included in the short exact sequence of bundles on Gr(E_{x}, n) × S:

0 → G → p_{2}^{∗}E → A → 0, (8)

where p_{1}, p_{2} are the projections, A = p_{1}^{∗}Q⊗p_{2}^{∗}ℂ_{x} and Q is the universal quotient bundle on the Grassmanian.

Applying RHom_{p_{1}}(·, G) to (8) one obtains a long exact sequence of sheaves on the Grassmanian:

0 → Hom_{p_{1}}(G, G) → Ext_{p_{1}}^{1}(A, G) → Ext_{p_{1}}^{1}(p_{2}^{∗}E, G) →
→ Ext_{p_{1}}^{1}(G, G) → Ext_{p_{1}}^{2}(A, G) → 0.

Indeed, Hom_{p_{1}}(p_{2}^{∗}E, G) is zero since its fiber over F ∈ Gr(E_{x}, n) is Hom_{O_{S}}(E, F) which is zero (E and F are stable and c_{2}(F) > c_{2}(E)). We can put zero on the end because, due to the Technical Condition of Section 1, either both Ext_{p_{1}}^{2}(p_{2}^{∗}E, G) and Ext_{p_{1}}^{2}(G, G) are zero (when c_{1}(K) · c_{1}(H) < 0) or the map between them is an isomorphism (when K ≃ O). Moreover, Hom_{p_{1}}(G, G) is a trivial line bundle, since its fiber F is Hom_{O_{S}}(F, F) and stable sheaves have only scalar automorphisms.

Thus, the full Chern class of T_{M^{G}(r,n+s)} is equal to:

c(T_{M^{G}(r,n+s)}) = c(Ext_{p_{1}}^{1}(G, G)) = \frac{c(Ext_{p_{1}}^{2}(A, G))c(Ext_{p_{1}}^{1}(p_{2}^{∗}E, G))}{c(Ext_{p_{1}}^{1}(A, G))}.

Now we proceed to compute the three terms in the formula above.

To compute c(Ext_{p_{1}}^{1}(p_{2}^{∗}E, G)) we apply RHom_{p_{1}}(p_{2}^{∗}E, ·) to (8) and get

0 → Hom_{p_{1}}(p_{2}^{∗}E, p_{2}^{∗}E) → Hom_{p_{1}}(p_{2}^{∗}E, A) →
→ Ext_{p_{1}}^{1}(p_{2}^{∗}E, G) → Ext_{p_{1}}^{1}(p_{2}^{∗}E, p_{2}^{∗}E) → 0.

Here, the sheaf Hom_{p_{1}}(p_{2}^{∗}E, G) vanishes as before and Ext^{1}(p_{2}^{∗}E, A) is zero since its fiber over a point F is Ext_{O_{S}}^{1}(E, (ℂ_{x})^{n}) = H^{1}(S, E^{∗} ⊗ (ℂ_{x})^{n}) = 0. The bundles Hom_{p_{1}}(p_{2}^{∗}E, p_{2}^{∗}E) and Ext_{p_{1}}^{1}(p_{2}^{∗}E, p_{2}^{∗}E) are trivial on Gr(E_{x}, n) hence

c(Ext_{p_{1}}^{1}(p_{2}^{∗}E, G)) = c(Hom_{p_{1}}(p_{2}^{∗}E, A)) = c(Q ⊗ ℂ^{r}) = (c(Q))^{r}.

To compute c(Ext_{p_{1}}^{1}(A, G)) we apply RHom_{p_{1}}(A, ·) to (8) and get

0 → Hom_{p_{1}}(A, A) → Ext_{p_{1}}^{1}(A, G) → 0 → Ext_{p_{1}}^{1}(A, A) →
→ Ext_{p_{1}}^{2}(A, G) → Ext_{p_{1}}^{2}(A, p_{2}^{∗}E) → Ext_{p_{1}}^{2}(A, A) → 0

where Ext_{p_{1}}^{1}(A, p_{2}^{∗}E) vanishes since it is Serre dual to Ext_{p_{1}}^{1}(p_{2}^{∗}E, A ⊗ p_{2}^{∗}K_{S}) (cf. [19]), and the fibers of the latter sheaf compute first cohomology of an Artin sheaf. Since Hom_{p_{1}}(A, A) ≃ Q ⊗ Q^{∗}, we conclude that c(Ext_{p_{1}}^{1}(A, G)) = c(Q ⊗ Q^{∗}).

To compute c(Ext_{p_{1}}^{2}(A, G)) we use the exact sequence from the computation of c(Ext_{p_{1}}^{1}(A, G)). From Ext_{O_{S}}^{2}(ℂ_{x}, ℂ_{x}) = ℂ and Ext_{O_{S}}^{1}(ℂ_{x}, ℂ_{x}) = ℂ^{2}, we deduce

Ext_{p_{1}}^{2}(A, A) ≃ Q ⊗ Q^{∗}, Ext_{p_{1}}^{1}(A, A) = (Q ⊗ Q^{∗})^{⊕2}.

therefore, it only remains to compute Ext_{p_{1}}^{2}(A, p_{2}^{∗}E). To that end, we again apply a relative version of Serre’s duality (loc. cit.) which gives

Ext_{p_{1}}^{2}(A, p_{2}^{∗}E) ≃ (Q^{∗})^{⊕r}.

and hence c(Ext_{p_{1}}^{2}(A, G)) = c(Q ⊗ Q^{∗})(c(Q^{∗}))^{r}.

Summing up the results of our computation, we obtain:

c(T_{M^{G}(r,n+s)}) = \frac{c(Q ⊗ Q^{∗})(c(Q^{∗}))^{r} · (c(Q))^{r}}{c(Q ⊗ Q^{∗})} = (c(Q)c(Q^{∗}))^{r}.

q.e.d.

A similar approach can be used with T_{Quot_{C}^{n}(E)} and T_{Quot_{C′}^{n}(s)} (one has to consider sheaves on Gr(E_{x}, n)×C and Gr(E_{x}, n)×C′). This leads to:

Lemma 6.2. The following equalities for the full Chern classes of sheaves on Gr(E_{x}, n) hold:

c(T_{Quot_{C}^{n}(E)}) = c(T_{Quot_{C′}^{n}} (s)) = (c(Q))^{r},

where c(Q) is the full Chern class of the universal quotient bundle.

Let S be the universal subbundle on Gr(E_{x}, n). Then T_{Gr(E_{x},n)} ≃ S^{∗} ⊗ Q. Now using the exact sequence (7) we obtain

c(V) = \frac{c(T_{M^{G}(r,n+s)})c(T_{Gr(E_{x},n)})}{c(T_{Quot_{C}^{n}(E)})c(T_{Quot_{C′}^{n}(s)})} = \frac{(c(Q)c(Q^{∗}))^{r}(c(S^{∗} ⊗ Q))}{(c(Q))^{r}(c(Q))^{r}}
= \frac{(c(Q^{∗}))^{r}(c(S^{∗} ⊗ Q))}{(c(Q))^{r}} = \frac{(c(Q^{∗}))^{r}}{c(Q ⊗ Q^{∗})}
= c(Q^{∗} ⊗ S).

The equality in the second line is obtained by applying (…) ⊗ Q to the dual of the tautological exact sequence 0 → S → O^{r} → Q → 0. The last equality is obtained by applying (…) ⊗ Q^{∗} to the tautological sequence itself.

Note that S^{∗} ⊗ Q is the cotangent bundle to the Grassmanian. Its top Chern class (which can be computed from the topological Euler characteristic of Gr(E_{x}, n)) is equal to (−1)^{(r−1)n}\binom{r}{n}. Recall that this number computes the intersection index of the cycles [Quot_{C}^{n}(E)] and [Quot_{C′}^{n}(s)] in the case when C and C′ are a pair of distinct lines on ℙ^{2}. In the general case, the curves C and C′ will intersect in ⟨C, C′⟩ points. Then we have the

Corollary 6.3.

∑_{n=1}^{∞} z^{2n}⟨Quot_{C}^{n}(E), Quot_{C′}^{n}(s)⟩ = (1 + (−1)^{r−1}z^{2})^{r⟨C,C′⟩}

Proof. Denote ⟨C, C′⟩ by q and (−1)^{(r−1)n}\binom{r}{n} by a_{n}. It follows from the definitions and the discussion in the beginning of this chapter, that the set-theoretic intersection Quot_{C}^{n}(E) ∩ Quot_{C′}^{n}(s) corresponds to sheaves F which fit into a short exact sequence 0 → F → E → A, such that A is actually a quotient of ⨁_{x∈C∩C′} E_{x}. The variety of such

sheaves has several irreducible components according to the multiplicity of A at the points on C ∩ C′. Thus, each component is labeled by a subdivision of n into a sum n_{1} + … + n_{q} of q ordered nonnegative integers. Such a component is a product of the Grassmanians ∏_{x_{i}∈C∩C′}Gr(E_{x}, n_{i}), and it has the excess intersection bundle as before.

It is easy to show that the top Chern class of this bundle will be equal to the product a_{n_{1}} · . . . · a_{n_{q}} (since all data will split into direct product over the individual Grassmanians). Thus, to get the intersection number ⟨Quot_{C}^{n}(E), Quot_{C′}^{n}(s)⟩ we need to sum up the products a_{n_{1}} ·. . .·a_{n_{q}} over all subdivisions n = n_{1} + … + n_{q}. Since a_{k} = (−1)^{(r−1)k}\binom{r}{k}, a standard combinatorial argument and the binomial formula show that

∑_{n=1}^{∞} z^{2n}⟨Quot_{C}^{n}(E), Quot_{C′}^{n}(s)⟩ = (1 + (−1)^{r−1}z^{2})^{r⟨C,C′⟩}. q.e.d.

A. Appendix: The punctual quot scheme

In this appendix we outline the proof of Theorem 2.2 which claims that the punctual Quot scheme Quot(r, n) is irreducible of dimension rn − 1. The full details can be found in [2].

Since the main object of interest in this appendix is local in its nature, we will assume that the surface S is ℂ^{2}. Recall that the closed points of Quot(r, n) are in bijective correspondence with all length n Artin quotients O_{ℂ^{2}}^{r} → A which are supported at the origin. This allows us to give a description of Quot(r, n) in terms of linear algebra, generalizing Nakajima’s construction for Hilbert schemes, cf. [27].

Let A be such a quotient and choose an isomorphism of H^{0}(ℂ^{2}, A) with a standard n-dimensional vector space V. Then multiplication by the two coordinate functions on ℂ^{2} gives two commuting operators B_{1} and B_{2} on V. It is easy to see that A is supporeted at 0 ∈ ℂ^{2} if and only if B_{1} and B_{2} are nilpotent. Moreover, the r natural constant sections of O_{ℂ^{2}}^{r} map to certain vectors v_{1}, …, v_{r} in V which satisfy the following

Definition. Let V be a vector space with two operators B_{1}, B_{2}. A set of vectors v_{1}, …, v_{r} is called cyclic if there are no proper subspaces W ⊂ V, such that

(a) v_{i} ∈ W, i = 1, …, r,

(b) B_{1}(W) ⊂ W, B_{2}(W) ⊂ W.

Let N_{2} be the space of pairs of commuting nilpotent operators on V, a closed affine subvariety of End(V) ⊕ End(V).

Consider also a subspace U_{r} of N_{2} × V^{⊕r} formed by all (B_{1}, B_{2}, v_{1}, …, v_{r}), such that v_{1}, …, v_{r} is a cyclic set of vectors. It is easy to see that U_{r} is an open subset in N_{2} × V^{⊕r} (it is given by a condition saying that some system of vectors in V has maximal rank). The group GL(V) acts on U_{r} by conjugating the nilpotent operators and acting on vectors diagonally.

Lemma A.1. GL(V) acts freely on U_{r}.

Proof. Suppose g ∈ GL(V) stabilizes (B_{1}, B_{2}, v_{1}, …, v_{r}) ∈ U_{r}. Then Ker(1−g) contains v_{1}, …, v_{r}. Since it is also preserved by B_{1}, B_{2}, we have Ker(1 − g) = V and therefore g = 1. q.e.d.

The following lemma gives an explicit construction of the punctual Quot scheme:

Lemma A.2. There exists a morphism π : U_{r} → Quot(r, n) such that:

(i) π is surjective;

(ii) The fibers of π are precisely the orbits of GL(V) action on U_{r}.

Proof. Clear from the discussion preceding the definition of a cyclic set of vectors. q.e.d.

Our strategy in proving Theorem 2.2 is as follows. First we note that we have

Theorem A.3. U_{1} is irreducible of dimension n^{2} + n − 1.

Proof. Follows from the previous Lemma and the irreduciblity and dimension of Hilb^{n} = Quot(1, n) proved in [5], [18]. q.e.d.

Once we manage to show that U_{1} is dense in N_{2}, that will imply that N_{2} is irreducible of dimension n^{2} − 1. Since U_{r} is open in N_{2} × V^{⊕r}, by Lemma A.2 we will be able to prove the corresponding statements for Quot(r, n).

Lemma A.4 (cf. [2]). Let B_{1}, B_{2} be two commuting nilpotent operators on a vector space V. There exists a third nilpotent operator B′_{2} and a vector w ∈ V such that:

(i) B′_{2} commutes with B_{1};

(ii) Any linear combination αB_{2} + βB′_{2} is nilpotent;

(iii) w is a cyclic vector for the pair of operators (B_{1}, B′_{2}).

Proof. See Lemma 3 in [2]. q.e.d.

Theorem A.5. The subset U_{1} defined above is dense in N_{2} × V.

Proof. Let (B_{1}, B_{2}, v) be any point in N_{2} × V. Consider the triple (B_{1}, B′_{2}, w) ∈ U provided by the above lemma. Connect the two triples (B_{1}, B_{2}, v) and (B_{1}, B′_{2}, w) by an affine line L inside the vector space gl(V) ⊕ gl(V) ⊕ V. Due to the choice of B′_{2} the whole line L belongs to the closed subspace N_{2} × V ⊂ gl(V) ⊕ gl(V) ⊕ V. Since U ∩ L is open in L and non-empty, (B_{1}, B_{2}, v) belongs to the closure of U. q.e.d.

Corollary A.6. The punctual Quot scheme Quot(r, n) is irreducible of dimension rn − 1.

Proof. By the previous theorem N_{2} is irreducible of dimension n^{2} − 1. Since U_{r} is dense in N_{2} × V^{⊕r}, it is irreducible of dimension n^{2} −1+rn. Finally, by Lemma A.2 the variety U_{r} is a principal GL(V)- bundle over Quot(r, n) therefore Quot(r, n) is irreducible of dimension rn − 1. q.e.d.

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