Moduli of Sheaves on Surfaces and Action of the Oscillator Algebra

Vladimir BARANOVSKY^{1}

Department of Mathematics, University of Chicago 5734 S.University ave., Chicago, IL 60637, USA

Introduction

Let S be a smooth complex projective surface and Hilb^{n}(S) the Hilbert scheme of all length n zero-dimensional subschemes of S. It is known (cf. [Fo]) 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)) (all cohomology in this paper will be with complex coefficients) the picture becomes more comprehensible.

Firstly, for any complex smooth algebraic variety X of dimension d, let P_{t}(X) be 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

∑_{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 [VW] have noticed that the right hand side of the formula above is an irreducible character of the oscillator (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) [p_{i}^{α}, p_{j}^{β}] = ⟨α, β⟩i · δ_{i+j,0}; where the commutator above is to be understood in the graded sense and ⟨·, ·⟩ is the intersection form on S. The action of the oscillator algebra can be interpreted as follows: there is a standard realization of Fock representation in the space of all symmetric polynomials in infinitely many variables. This realization appears from the cohomology of symmetric powers Sym^{n}(S) which are related to Hilbert schemes of points via the Hilbert-Chow morphism Hilb^{n}(S) → Sym^{n}(S).

1email: barashek@math.uchicago.edu

And indeed, Nakajima and Grojnowski (cf. [Na], [Gr]) have constructed the expected action of H on ⨁_{n≥0} H^{∗}(Hilb^{n}(S), ℂ) using some explicit 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 [VW] (see also [Na], [Gr]) 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 [Ku]), instantons on ALE spaces (quiver variety and the Uhlenbeck-type compactification cf. [N]), 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. [Br], [Ia]) was also proved independently by Ellingsrud and Lehn, cf. [EL]). This leads to the second important theorem saying that, in the coprime case, the natural morphism M^{G}(r, n) → M^{U}(r, n) is strictly semismall 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 of Uhlenbeck compactifications have recently appeared in string theory. Since L^{2}- cohomology coincides in several interesting cases with intersection homology, this provides another motivation for studying these homology groups. This paper can be regarded as a first step in application of intersection homology to moduli spaces of sheaves on surfaces.

Acknowledgements. The author thanks his advisor V. Ginzburg for stating the problem and for his valuable comments and support. The author is also grateful to L. Göttsche, A.S. Strømme and Z. Qin for their useful advices and remarks. Finally, the author thanks Max-Planck Institut für Mathematik, where this work was completed, for its hospitality and excellent research conditions.

Contents

1 Moduli spaces. 3

2 Gieseker to Uhlenbeck. 4

3 Stratifications and semi-small maps. 7

4 Correspondences. 13

5 A transversality result. 20

6 Computation of the intersection number. 24

7 Appendix: the punctual Quot scheme. 28

1 Moduli spaces.

Let S be a smooth complex projective surface. We denote by Sym^{n}S the n-th symmetric power of S. Choose and fix an ample line bundle H, an arbitrary line bundle L and a positive integer r.

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 (cf. [Ma]). We will only consider the case when gcd(r, c_{1}(L) · c_{1}(H)) = 1. Since the bundles L and H will be fixed we drop them from notation.

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. [Ma]). The second compactification is the Uhlenbeck moduli space M^{U}(r, n) which can be described as follows. Take the disjoint union ∐_{s=0}^{∞} N(r, n − s) × Sym^{s}S. Then Uhlenbeck’s theorem [Uh] 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. Following [DK] one can show that this topology is even metrizable. This topological space is called the Uhlenbeck compactification M^{U}(r, n) of N(r, n).

Note that the union above is in fact finite since a necessary condition for non-emptiness of N(r, k) is the Bogomolov inequality 2rk−(r−1)(c_{1}(L))^{2} > 0. We can always tensor all our bundles and sheaves with a high power of H and achieve c_{1}^{2}(L) > 0. Then Bogomolov inequality implies that k = n − s should be at least positive.

Hence without loss of generality we can assume that 0 ≤ s < 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), some components may be of dimension higher than expected, etc. For this reason we introduce a

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 ensure (cf. [HL]) that for any n ≥ 0 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}). (1)

This condition (b) is automatic when (−K) is represented by an effective curve. Note that since N(r, n) is an open subset of M^{G}(r, n) (loc. cit.), it also has to be 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. [DK 10.3.4]) that for some elliptic surfaces or surfaces of general type the condition (1) may fail.

2 Gieseker to Uhlenbeck.

The Gieseker and Uhlenbeck compacifications are higher rank analogues of the Hilbert scheme of all length n subschemes Hilb^{n}(S) and the symmetric power Sym^{n}(S), respectively. Here we prefer to think of Hilb^{n}(S) as parametrizing rather the ideal sheaves, i.e. torsion-free sheaves of rank one with c_{2} = n and trivial determinant. One has a natural map a_{1} : Hilb^{n}(S) → Sym^{n}(S):

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

where J_{ξ} is an ideal sheaf of a subscheme ξ and the formal sum above is viewed as an element of Sym^{n}(S).

There exists a natural map a_{r} : M^{G}(r, n) → M^{U}(r, n) generalizing a_{1}. To define a_{r} first note that, for any torsion-free sheaf F, the double-dual F^{∗∗} is reflexive. It is known (cf. [OSS]) that any reflexive sheaf on a smooth variety is locally free on the complement of a closed subset of codimension ≥ 3. Hence in our case (dim S = 2) the double-dual is necessarily locally free.

Consider the short exact sequence:

0 → F → F^{∗∗} → A_{F} → 0.

The quotient sheaf A_{F} = F^{∗∗}/F is supported on finitely many points (i.e. A_{F} is an Artin sheaf). Denote by l_{F} the length of A_{F}. We define a_{r}(F) 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 moduli space M^{G}(r, n) parametrizes only S-equivalence classes (cf. [Ma]) of semistable sheaves and one has to be more careful to make sure that a_{r} is well-defined on such classes (cf. [Li]).

It takes quite a lot of technical work (see [Li], [Mo]) to show that the map a_{r} is continuous (this is because one has to use gauge-theoretic Uhlenbeck Compactness Theorem and consider M^{U}(r, n) as a topological space). We will not attempt to repeat the argument here referring the interested reader to the references cited. Moreover, from now on we assume that M^{U}(r, n) has a structure of a projective algebraic variety and a_{r} is algebraic. In rank 2 case this was proved in [Li] and we intend to give an alternative construction for the general case in a forthcoming paper.

We need a description of 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 and we drop x from notation. 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. [Ba] and [EL]):

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

Proof. See Appendix. □

The first application of Theorem 2.2 is to show that, under the technical assumption above, the moduli spaces M^{G}(r, n) and M^{U}(r, n) are wellbehaved.

Theorem 2.3 Assume that the 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. □

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 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} with x_{i} pairwise distinct (veiwed as elements of Sym^{n}S). 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. [BM] or [CG, 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. [Br], [Ia]).

Proposition 3.1 (cf. [GS])

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 [GS] this will lead to some formulas for Poincaré polynomials. These formulas will be later interpreted in representation-theoretic 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 1 the fiber is isomorphic to ∏_{i=1}^{m} Quot(r, m_{i}) and by Theorem 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). □

The semi-smallness result will allow us to relate homological invariants of M^{G}(r, n) and M^{U}(r, n). Recall (cf. [BBD]) 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. [BBD] of [GS]).

Following [Sa, 1.13] one can define pure Hodge structure on the intersection homology of any complex algebraic variety X.

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. [BBD]).

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 ICcomplex 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. □

Now [GS, 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}.

Similarly, 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.

With this preparation, we can deduce the first main result of the paper.

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 [Mc1] 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)}} . □

Remarks.

(1) A very similar formula was discovered by Göttsche in [Gö22]. 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 [LQ] 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.

Example. When S = ℙ^{2} and r = 2 the generating function for M^{G}(r, n) was computed by Yoshioka in [Y]. In this case

∑_{i=0}^{∞} q^{n}P_{t}(M^{G}(r, n)) =

= \frac{∑b∈ℤ \frac{t^{−2b}q^{b^{2}}}{1−t^{4}q^{2b−1}}}{∑_{n∈ℤ} t^{−2n}q^{n^{2}}} ·\frac{1}{t^{4}(t^{2} − 1) ∏_{l=1}^{∞}(1 − t^{−2}q^{l})^{2}(1 − q^{l})^{2}(1 − t^{2}q^{l})^{2}}

Let q = e^{2πiτ} and t = e^{2πiz}. Recall the classical theta functions:

θ_{μ,ν}(τ, z) := ∑_{n∈ℤ}(−1)^{nν}q^{(n+μ/2)^{2}/2}t^{n+μ/2} (μ, ν ∈ {0, 1}).

Then by the product formula

θ_{1,1}(τ, z) = q^{1/8}(t^{1/2} − t^{−1/2}) ∏_{l>0}(1 − t^{−1}q^{l})(1 − q^{l})(1 − tq^{l})

for theta functions one can write

∑_{i=0}^{∞} q^{n}P_{t}(M^{G}(r, n)) = ∑_{b∈ℤ} \frac{t^{−2b}q^{b^{2}}}{1 − t^{4}q^{2b−1}} · \frac{q^{1/4}(t − t^{−1})}{t^{5} θ_{0,0}(2τ, 2z)θ_{1,1}(τ, 2z)^{2}}

By the Theorem above we immediately deduce that

∑_{i=0}^{∞} q^{n}P_{t}(M^{U}(r, n)) = ∑_{b∈ℤ} \frac{t^{−2b}q^{b^{2}}}{1 − t^{4}q^{2b−1}} · \frac{q^{1/8}}{t^{5} θ_{0,0}(2τ, 2z)θ_{1,1}(τ, 2z)}

Remark. Conjecturally, the theta functions in the generating series above appear from extra symmetries of cohomology of the moduli spaces.

Using Saito’s theory of mixed Hodge modules (cf. [Sa]) and the formula for Hodge polynomials of symmetric powers (cf. [Bu]) one can apply 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 (shifted) intersection homology Hodge polynomials of M^{G} and M^{U}:

\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} ∏_{i,j=−1}^{1}(1 + (−1)^{i+j+1}x^{i}y^{j}q^{l})^{(−1)^{i+j+1}h^{i+1,j+1}(S)};

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

Remark. 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.

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 vector can be naturally identified with intersection homology of Uhlenbeck compactifications.

Recall [CG] that, for any two smooth projective varieties M_{1}, M_{2}, a closed subvariety Z ⊂ M_{1} × M_{2} (more generally, any cohomology class in H^{∗}(M_{1} × M_{2})), defines a map [Z] : H^{∗}(M_{1}) → H^{∗}(M_{2}) given by [Z](α) = (p_{2})_{∗}([Z] ∩ (p_{1})^{∗}), where p_{1} and p_{2} are the projections onto M_{1} and M_{2}, respectively. If M_{3} is a third smooth projective variety and Z′ ⊂ M_{2} × M_{3} then the composition [Z′] ∘ [Z] : H^{∗}(M_{1}) → H^{∗}(M_{3}) can be described as follows. Let p_{ij} : M_{1} ×M_{2} ×M_{3} → M_{i} ×M_{j} be the projection to the product of the i-th and j-th factors. Then [Z′] ∘ [Z] is given by the cohomology class (p_{13})_{∗}(p_{12}^{∗}[Z] ∪ p_{23}^{∗}[Z′]) ∈ H^{∗}(M_{1} × M_{3}).

We want to apply this construction by introducing certain correspondences in the products of moduli spaces.

Definition. For i > 0, define a subvariety P_{−i} ⊂ ∐_{n=0}^{∞} M^{G}(r, n) × M(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.

Note that the second condition makes sense since the double dual is stable and therefore simple.

Similarly, we define P_{i} for i > 0, exchanging F_{1} and F_{2}. Let us define a morphism Π : P_{i} → S 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 [HL, 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. (1.1). 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 [Na 8.3] and Theorem 2.2).

Let P_{±i}^{α} ∈ ∏_{n} H^{∗}(M^{G}(r, n ∓ i)) ⊗ H^{∗}(M^{G}(r, n)), α ∈ H^{∗}(S), be the images of the cohomology classes Π^{∗}α on P_{±i}. The cycles P_{±i}^{α} induce the maps ⨁_{n} H^{∗}(M^{G}(r, n)) → ⨁_{n} H^{∗}(M^{G}(r ∓ i)) via the convolution construction given above. Computing dimensions we can see that P_{−i}^{α} raises cohomological degree by 2(ri − 1) + deg α.

The next theorem asserts that ⨁_{n=0}^{∞} H^{∗}(M^{G}(r, n)) becomes a representation 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 [Na, Theorem 8.13].

Theorem 4.1 The following relations hold,

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

where ⟨α, β⟩ is the intersection form on S.

Exactly as in [Na Corollary 8.16] we deduce a

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 as follows.

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. [OSS, 3.2]). The Uhlenbeck compactification M^{U}(2, 2) can be described as follows. Let V be the 6-dimensional 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. [OSS, 4.3]). Its singular locus P ≃ ℙ^{2} corresponds to symmetric matrices of rank one. It can be deduced from (cf. [OSS, 4.3]) that M^{G}(2, 2) coincides with 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] be the generator of H^{0}(M^{G}(2, 1)) = H^{0}(pt). 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.

Beginning of the proof of 4.1: Most of Nakajima’s proof 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 [Na 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. □

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 formalizm 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} from 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}. □

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). In particular, the subspace ⨁ H^{rn}(M^{G}(r, n)) is preserved by P_{−i}^{C}. Moreover, the next theorem shows that in fact even the subspace generated by the classes of L^{μ,s}, is invariant under P_{−i}^{C} (i > 0).

Theorem 4.6 (cf. [Na, Theorem 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. [Na, 9.1], [Mc2]). 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 [Na, formula 9.16]) we obtain the formulas

∑_{n=0}^{∞} z^{n}[Quot_{C}^{n}(E)] = exp( ∑_{i=1}^{∞} \frac{z^{i}P_{i}^{C}}{(−1)^{i−1}i}) · [vac].  (2)

∑_{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. [Mc2]). They are the first ingredient in our computation of c_{i}. The second ingredient is provided by Theorem 4.7 (cf. [Na, 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).

End of 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 ⨁_{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 (2) 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.

For any pair of operators A and B, one has an identity

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

Therefore

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

5 A transversality result.

In r = 1 case Theorem 4.6 has a simple proof (cf. [Na]) 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 next step 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.

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

Lemma 5.1 Let (F_{1}, F_{2}) ∈ P_{−i}^{C} ⋂ M^{G}(s + |μ|) × M^{G}(s + |λ|) be as 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. [Ar]) 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 decom-

position

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 before this lemma. It is easy to show using this local description that 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}}).

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

Step 2. Recall that any tangent vector v ∈ T_{F_{1}}(M^{G}(r, s + |μ|)) ⊂ Ext_{S}^{1}(F_{1}, F_{1}) corresponds to an extension 0 → F_{1} → G_{1} → F_{1} → 0 (i.e. deformation with base ℂ[ϵ]/ϵ^{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}  (3)

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

The diagram (3) 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) ∈ U and v ∈ T_{F_{1}}(L^{μ,s}) then w ∈ T_{F_{2}}(L^{λ,s}) and 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}  (4)

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 (4) and

the upper row comes brom applying Hom_{S}(F_{2}, · ) to the upper row of (4). 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 chase 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 if v ∈ Hom_{S}(F_{1}, A_{F_{1}}) ⊂ Ext^{1}(F_{1}, F_{1}) then 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 completes the proof of the lemma and hence the proof of Theorem 4.6. □

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 trasversal 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. [Fu]).

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). Since E_{1} is a rank r vector bundle on C this space is of constant dimension rn.

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 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, 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. [Fu, 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. (5)

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 this Chern class we will need to consider certain sheaves on M^{G}(r, n + s) × S. 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 [Kl] Ext_{p_{1}}^{i}(G_{1}, G_{2}) is a vector bundle on M^{G}(r, n + s) of the same rank. 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).

Consider a larger moduli space M̃^{G}(r, n + s) of H-stable sheaves with c_{1}(F) = c_{1}(L) and c_{2}(F) = n + s (i.e. the determinant is not fixed but is allowed to differ from L by any line bundle in P ic^{0}(S)). Since gcd(r, c_{1}(H) · c_{1}(L)) = 1, there exists a universal sheaf G on M̃^{G}(r, n+s)×S (cf. [HL]). It is known (cf. [Ma]) that the tangent bundle T M̃^{G}(r, n + s) is isomorphic to Ext_{p_{1}}^{1}(G, G). Since M^{G}(r, n + s) is a fiber of a smooth map det : M̃^{G}(r, n + s) → P ic^{0}(S), the tangent bundle to M^{G}(r, n + s) has the same Chern classes as Ext_{p_{1}}^{1}(G, G).

To compute the full Chern class of Ext_{p_{1}}^{1}(G, G)|_{Gr(E_{x},n)} we use a short exact sequence of sheaves on Gr(E_{x}, n) × S:

0 → G → p_{2}^{∗}E → A → 0,  (6)

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.

First of all, applying RHom_{p_{1}}(·, G) to (6) 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

Note that Hom_{p_{1}}(p_{2}^{∗}E, G) vanishes 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)). Moreover, the fiber of Hom_{p_{1}}(G, G) over F is Hom_{O_{S}}(F, F). Since stable sheaves have only scalar automorphisms, Hom_{p_{1}}(G, G) is a trivial line bundle. One can also check that 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).

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))}

As a second step we apply RHom_{p_{1}}(p_{2}^{∗}E, ·) to (6) 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}.

Finally, we apply RHom_{p_{1}}(A, ·) to (6) 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 dual to Ext_{p_{1}}^{1}(p_{2}^{∗}E, A ⊗ p_{2}^{∗}K_{S}) (cf. [Kl]). Since Hom_{O_{S}}(ℂ_{x}, ℂ_{x}) = Ext_{O_{S}}^{2}(ℂ_{x}, ℂ_{x}) = ℂ and Ext_{O_{S}}^{1}(ℂ_{x}, ℂ_{x}) = ℂ^{2}, we have

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

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

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

Summing up the results of our computation, we obtain

c(T_{M^{G}(r,n+s)}) = \frac{(c(Q^{∗}))^{r}(c(Q))^{r}(c(Q ⊗ Q^{∗}))^{2}}{c(Q ⊗ Q^{∗})c(Q ⊗ Q^{∗})} = (c(Q)c(Q^{∗}))^{r}.

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′). One checks that

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

Let S be the universal subbundle on Gr(E_{x}, n). Using the exact sequence (5) and the following short exact sequence:

0 → Q ⊗ Q^{∗} → Q^{⊕r} → S^{∗} ⊗ Q → 0

(recall that T_{Gr(E_{x},n)} ≃ S^{∗} ⊗ Q), we get

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(Q))^{r}}{(c(Q))^{r}c(Q ⊗ Q^{∗})} = \frac{(c(Q^{∗}))^{r}}{c(Q ⊗ Q^{∗})} = c(Q^{∗} ⊗ S).

Hence the full Chern class c(V) of the excess normal bundle V is equal to the full Chern class c(Q^{∗}⊗S) of the cotangent bundle to Gr(E_{x}, n) (which is isomorphic to Q^{∗} ⊗ S). Therefore

c_{(r−n)n}(V) = (−1)^{rn−n^{2}} \binom{r}{n} = (−1)^{(r−1)n}\binom{r}{n}.

Recall that the top Chern class above is equal to the intersection number of [Quot_{C}^{n}(E)] and [Quot_{C′}^{n}(s)]. Now Theorem 4.7 follows from the binomial formula. □

7 Appendix: the punctual Quot scheme.

In this appendix we give a proof of Theorem 2.2 saying that the scheme Quot(r, n) is irreducible of dimension (rn − 1).

Our strategy is to find a dense irreducible open subset W ⊂ Quot(r, n) of dimension (rn − 1). This subset will turn out to be a vector bundle over the punctual Hilber scheme Hilb^{n} := Quot(1, n) considered by Briançon and Iarrobino.

We define W as the set of all quotients O^{⊕r} ⟶^{ϕ} A, ϕ = (ϕ_{1}+ϕ_{2}+…+ϕ_{r}) such that the first component ϕ_{1} : O → A is surjective (this is clearly an open condition). Such a ϕ_{1} corresponds to a point in Hilb^{n}. Once ϕ_{1} is chosen, the other components (ϕ_{2}, …, ϕ_{r}) are given by an arbitrary element of Hom(O^{⊕(r−1)}, A) = ℂ^{(r−1)n}. Therefore W is a rank (r−1)n vector bundle over Hilb^{n}. By [Br] or [Ia] the subset W is irreducible of dimension (rd−1).

Now we want to show that W is dense in Quot(r, n). In fact, for any point x ∈ Quot(r, n) we will find an irreducible rational curve C ⊂ Quot(r, n) connecting it with some point in W.

To that end, we generalize Nakajima’s construction (cf. [Na, Chapter 2]) of the global Hilbert scheme Hilb^{n}(ℂ^{2}) to the Quot scheme. Once we do that, the existence of the irreducible curve will amount to an exercise in linear algebra (cf. Lemma 7.3).

Fix a complex vector space V of dimension n, and N_{n} let be the space of pairs of commuting nilpotent operators on V. The space N_{n} is naturally a closed affine subvariety of End(V) ⊕ End(V).

Consider a subspace U_{r} of N_{n} × V^{⊕r} formed by all (B_{1}, B_{2}, v_{1}, …, v_{r}), such that there is no proper subspace of V which is invariant under B_{1}, B_{2} and contains v_{1}, …, v_{r}. Then U_{1} × V^{⊕(r−1)} ⊂ U_{2} × V^{⊕(r−2)} ⊂ … ⊂ U_{r} is a chain of open subsets in N_{n} × V^{⊕r} (each of them is given by a condition saying that some system of vectors in V has maximal rank).

Ona has a natural GL(V)-action on V_{r} and it is easy to prove that U_{r} is GL(V)-stable.

Lemma 7.1 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. □

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

Lemma 7.2 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}; (iii) π^{−1}(W) = U_{1} × V^{⊕(r−1)}.

Proof. Note that the punctual Quot scheme does not depend on the surface since the n-th power of the maximal ideal 𝔪_{x} acts by zero on any length n sheaf supported at x. Hence we can assume that S = ℂ^{2} = Spec ℂ[x_{1}, x_{2}] and that all the quotients are supported at x = 0 ∈ ℂ^{2}.

To construct π suppose that (B_{1}, B_{2}, v_{1}, …, v_{r}) is a point in U_{r} and consider a ℂ[x_{1}, x_{2}]-module structure on V in which x_{1} acts by B_{1} and x_{2} acts by B_{2}. We can view V as a quotient of a free ℂ[x_{1}, x_{2}]-module with generators v_{1}, …, v_{r}. Since B_{1} and B_{2} are nilpotent \sqrt{Ann(V)} = (x_{1}, x_{2}). Therefore a coherent sheaf A on ℂ^{2} associated with V is a quotient of O^{⊕r} supported at s. Moreover, V ≃ H^{0}(S, A) as vector spaces.

A different point in the same GL(V)-orbit defines an isomorphic quotient, hence the fibers of π : U_{r}/GL(V) → Quot(r, n) are GL(V)-invariant. Moreover, suppose that two points u_{1}, u_{2} of U_{r} give rise to isomorphic quotients A_{1}, A_{2}. Then the induced isomorphism between H^{0}(S, A_{1}) and H^{0}(S, A_{2}) defines an element of GL(V) taking u_{1} to u_{2}. Therefore, each fiber of π is precisely one GL(V)-orbit. This proves (ii).

To prove (i), suppose we have a quotient O^{⊕r} → A → 0 of length d supported at zero. Multiplication by x_{1} and x_{2} induces a pair of commuting nilpotent operators on H^{0}(S, A). Choose a ℂ-linear isomorphism H^{0}(S, A) ≃ V. The generators of the free ℂ[x_{1}, x_{2}]-module H^{0}(S, O^{⊕r}) project to some vectors v_{1}, …, v_{r} in V. Since v_{1}, …, v_{r} generate V as a ℂ[x_{1}, x_{2}]-module, (x_{1}, x_{2}, v_{1}, … v_{r}) is a point of U_{r}. Thus (i) is proved.

Finally, (iii) follows from definitions of W and U_{1}. □

Now we want to show that any point in U_{r} can be deformed to a point in the preimage of W. The above construction will allow us to construct this deformation using the following lemma

Lemma 7.3 Let B_{1}, B_{2} be two commuting nilpotent operators on a vector space V. There exists a third nilpotent operatov 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) (B_{1}, B′_{2}, w) ∈ U_{1}, i.e. w is a cyclic vector for the pair of operators (B_{1}, B′_{2}).

This lemma will be proved later. Now we will show how it can be used to give an

End of Proof of Theorem 2.2:

Let x be a point of Quot(r, n) and u_{1} = (B_{1}, B_{2}, v_{1}, …, v_{r}) be any point of π^{−1}(x) ⊂ U_{r}. Choose a nilpotent operator B′_{2} and a vector w ∈ V as in the lemma above. Connect the points u_{1} and u_{2} = (B_{1}, B′_{2}, w, v_{2}, …, v_{r}) with a straight line Φ(t), t ∈ ℂ such that Φ(1) = u_{1} and Φ(0) = u_{2}. This Φ(t) is given by equation:

Φ(t) = (B_{1}, tB′_{2} + (1 − t)B_{2}, tw + (1 − t)v_{1}, v_{2}, …, v_{r})

Note that for all t ∈ C, B_{2}(t) = tB′_{2} + (1 − t)B_{2} is nilpotent and commutes with B_{1}. Therefore the image of Φ(t) is a subset of N_{n} × V^{⊕r}. Since U_{r} is open in N_{n} × V^{⊕r}, there is a dense open subset C ⊂ ℂ such that Φ(C) ⊂ U_{r}. Similarly, there exists a dense open subset C_{1} ⊂ C such that Φ(C_{1}) ⊂ U_{1} × V^{⊕(r−1)}.

Hence the image π(Φ(C)) ⊂ Quot(r, n) is an irreducible rational curve connecting x = π(u_{1}) with π(u_{2}) ∈ W. Note that π(Φ(C_{1})) ⊂ W. Therefore x belongs to the closure of W. Since by Theorem 1.1 W is irreducible of dimension (rd − 1), the scheme Quot(r, n) is also irreducible of dimension (rd − 1). Theorem 2.2 is proved. □

Proof of Lemma 7.3:

Step 1. We will find a basis e_{i,j} of V, where 1 ≤ i ≤ k and 1 ≤ j ≤ μ_{i} such that

(a) B_{1}^{j−1}(e_{i,1}) = e_{i,j} for j ≤ μ_{i} and B_{1}^{μ_{j}}(e_{i,1}) = 0 (i.e. B_{1} has Jordan canonical form in the basis e_{i,j});

(b) B_{2}(e_{i,1}) ∈ ( ⨁_{k≥i+1} ℂ · e_{k,1}) ⊕ B_{1} · V.

To that end, recall one way to construct a Jordan basis for B_{1}. Let d = dim V and V_{i} = Ker(B_{1}^{d−i}). The subspaces V_{i} form a decreasing filtration V = V_{0} ⊃ V_{1} ⊃ V_{2} . . .. Moreover, B_{1} · V_{i} ⊂ V_{i+1}. Firstly, we choose a basis (w_{1}, …, w_{a_{1}}) of W_{1} := V_{0}/V_{1}. Lift this basis to some vectors e_{1,1}, e_{2,1}, …, e_{a_{1},1} in V_{0} and set all μ_{1}, …, μ_{a_{1}} equal to d. Secondly, choose a basis (w_{a_{1}+1}, …, w_{a_{2}}) of W_{2} := V_{1}/(B_{1} · V_{0} + V_{2}). Lift this basis to some vectors e_{a_{1}+1,1}, e_{2,1}, …, e_{a_{2},1} in V_{1} and set all μ_{a_{1}+1}, …, μ_{a_{2}} equal to d − 1. Continue in this manner by choosing bases of the spaces W_{i+1} = V_{i}/(B_{1} · V_{i−1} + V_{i+1}) and lifting them to V_{i}. This procedure gives us vectors e_{1,1}, e_{2,1}, …, e_{k,1} and the formula (a) tells us how to define e_{i,j} for j ≥ 2. It is easy to check that the system of vectors {e_{i,j}} is in fact a basis of V.

If we want to have property (b) we should be more careful with the choice of w_{i}. Note that all the subspaces V_{i} and B_{1} · V_{i} are B_{2}-invariant. Therefore we have an induced action of B_{2} on each W_{i}. We can choose our basis (w_{a_{i−1}+1}, …, w_{a_{i}}) of W_{i} in such a way that B_{2}(w_{i}) ∈ ⨁_{s=i+1}^{a_{i}} ℂ · w_{s} for all i ∈ {a_{i−1} + 1, …, a_{i}}. This ensures that (b) holds as well.

Step 2. Define B′_{2} by B′_{2}(e_{i,j}) = e_{i+1,j} if j ≤ μ_{i+1} and 0 otherwise. It is immediate that B′_{2} is nilpotent and that [B_{1}, B′_{2}] = 0. Let w = e_{1,1}. Then e_{i,j} = B_{1}^{j−1}(B_{2}^{i−1}(w)) hence (B_{1}, B′_{2}, w) ∈ U_{1}.

Step 3. Note that both B_{2} and B′_{2} are lower-triangular with zeros on the diagonal in the basis of V given by

e_{1,1}, e_{2,1}, …, e_{k,1}, e_{1,2}, e_{2,2}, … e_{k,2}, …

Hence any linear combination of B_{2} and B′_{2} is also lower-triangular and has zeros on the diagonal. Therefore αB_{2} + βB′_{2} is nilpotent for any complex α and β. The Lemma is proved. □

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