Universality of Descendent Integrals over Moduli Spaces of Stable Sheaves on $K3$ Surfaces

We interprete results of Markman on monodromy operators as a universality statement for descendent integrals over moduli spaces of stable sheaves on $K3$ surfaces. This yields effective methods to reduce these descendent integrals to integrals over the punctual Hilbert scheme of the $K3$ surface. As an application we establish the higher rank Segre-Verlinde correspondence for $K3$ surfaces as conjectured by G\"ottsche and Kool.


Descendent integrals
Let M be a proper and fine 1 moduli space of Gieseker stable sheaves F on a K3 surface S with Mukai vector v(F ) := ch(F ) td S = v ∈ H * (S, Z).
Let π M , π S be the projections of M × S to the factors and let F ∈ Coh(M × S) be a universal family. We define the k-th descendent of a class γ ∈ H * (S, Q) by τ k (γ) = π M * (π * S (γ)ch k (F)) ∈ H * (M ). (1.1)

Segre numbers
As a concrete application of Theorem 1.1 we prove a conjecture of Göttsche  For any topological K-theory class α ∈ K(S) define α M = ch −π M * π * S (α) ⊗ F ⊗ det(F) −1/ rk(v) whenever a rk(v)-th root of det(F) exists. Otherwise we define α M by a formal application of the Grothendieck-Riemann-Roch formula. Let c(α M ) be the Chern class corresponding to α M , see Remark 2.5.
For σ ∈ H * (S) consider, with the same convention if the root does not exist, the class µ M (σ) = −π M * ch 2 F ⊗ det(F) −1/ rk(v) π * S (σ) . We will usually drop the subscript M from the notation.
The Segre numbers of the Hilbert scheme of n points on the K3 surface S were determined by Marian, Oprea and Pandharipande [11]. In particular, they found the series V s , W s , X s . All that Theorem 1.2 does here is move their result from Hilbert schemes to moduli spaces of sheaves of arbitrary rank. Earlier work on Segre numbers can be found in [1,8,9,10,14].

Segre/Verlinde correspondence
Göttsche and Kool conjectured that the Segre numbers of moduli spaces of stable sheaves on surfaces are related by an explicit correspondence to the Verlinde numbers of these moduli spaces. For K3 surfaces the Verlinde numbers are known explicitly by and we refer to [3, equation (4)] for the definition of the class µ(L) ⊗ E ⊗r ∈ Pic(M ) Q . The Verlinde numbers of the Hilbert schemes of points of K3 surfaces (and in particular the series F r , G r ) were first computed in [1]. The computation for moduli spaces of higher rank sheaves reduces to the Hilbert scheme case as shown in [4] using hyperkähler geometry, parallel to Theorem 1.2. The functions F r , G r and V s , W s , X s are related by the following variable change [3]: Hence with Corollary 1.3 we have proven that the Segre and Verlinde numbers of moduli spaces of stable sheaves on K3 surfaces are related by this variable change. This is the K3 surface case of the higher-rank Segre-Verlinde correspondence conjectured by Göttsche

Plan
In Section 2, we use results from Markman's beautiful article [13] to formulate a universality result for descendent integrals of moduli spaces of stable sheaves on K3 surfaces, see Theorem 2.9. This immediately yields Theorem 1.1. In Section 3, we prove Theorem 1.2. where, if we decompose an element x ∈ Λ according to degree as (r, D, n), we have written x ∨ = (r, −D, n). We will also write Given a sheaf or complex E on S the Mukai vector of E is defined by Let v ∈ Λ be an effective 2 vector, H be an ample divisor on S and let be the moduli space of H-stable sheaves with Mukai vector v. The moduli space is smooth and holomorphic-symplectic of dimension 2 + (v, v). We further assume that the Mukai vector v is primitive, and the polarization H is v-generic (see [7,Theorem 6.2.5]), so that M is also proper (in particular, semistability is equivalent to stability). We also assume that there exists a universal sheaf F on M H (v) × S.
Remark 2.1. The results we state below also hold in the case where there exists only a twisted universal sheaf. More precisely, all statements below can be formulated in terms of the Chern character ch(F) alone and this class can be defined in the twisted case as well, see [12,Section 3]. The proofs carry over likewise since all ingredients hold in the twisted case as well.
Remark 2.2. More generally, one can also work with σ-stable objects for a Bridgeland stability condition in the distinguished component.
Assume from now on that 3 Consider the morphism θ F : where [−] k stands for taking the degree k component of a cohomology class. Then θ F restricts to an isomorphism which does not depend on the choice of universal family (use that the degree 0 component of the pushforward in (2.1) vanishes) and for which we hence have dropped the subscript F. The isomorphism θ is an isometry with respect to the Mukai pairing on the left, and the pairing given by the Beauville-Bogomolov-Fujiki form on the right. We will identify v ⊥ ⊂ Λ with The universal sheaf F and hence its Chern character ch(F) is uniquely determined only up to tensoring by the pullback of a line bundle from M . Following [13], we can pick a canonical normalization as follows: where we have suppressed the pullback by the projections to M and S in the first and last term on the right. We will follow similar conventions throughout. It is immediate to check that u v is independent from the choice of universal family (replace F by F ⊗ π * M L and calculate, see [13, Lemma 3.1]).
be the Hilbert scheme of n points on S. We have v = 1 − (n − 1)p, and we always take F = I Z , the ideal sheaf of the universal subscheme. If α ∈ H 2 (S) is the class of an effective divisor A ⊂ S, then is the class of the locus of subschemes incident to A. If we denote is the class of the locus of non-reduced subschemes, then under the identification (2.2) we have δ = − 1 + (n − 1)p . Because θ F (v) = −δ the canonical normalization of ch(F) takes the form

Markman's operator
Consider an isometry of Mukai lattices We identify g with an isometry . Hence the following diagram commutes Similar identification will apply to morphisms g defined over C. The Markman operator associated to g is given by the following result: Theorem 2.4 (Markman). For any isometry g : The operator is called the Markman operator and given by where π ij is the projection of M 1 × S 2 × M 2 to the (i, j)-th factor. Moreover, we have (c) γ(g 1 ) • γ(g 2 ) = γ(g 1 g 2 ) and γ(g) −1 = γ g −1 if it makes sense.
Remark 2.5. Here the Chern class c m in (2.3) has the following definition: Let be the universal map that takes the exponential Chern character to Chern classes, so in particular c(E) = (ch(E)) for any vector bundle. Then given α ∈ H * (M ) we write c m (α) for [ (α)] 2m .
Since γ(g) preserves degree and is isometric, it hence sends the class of a point on M 1 to the class of a point on M 2 . For any σ ∈ H * (M 1 ) we thus observe hat Proof of Theorem 2.4. If g is an integral isometry, then the statement of the theorem is a combination of Theorems 1.2 and 3.10 of [13]. The proof is involved: Markman establishes that operators γ(g) satisfying (a) and (b) exists by considering arbitrary compositions of parallel transport operators and pushforwards by isomorphisms induced by auto-equivalences. Then a small computation starting from an expression for the diagonal class of M 1 in terms of the universal sheaf F in [12], shows that conditions (a) and (b) for any homomorphism forces the expression (2.3). Hence those homomorphisms are uniquely determined. This last step holds even for homomorphisms defined over C which satisfy (a) and (b).
In the general case, one defines the operator γ(g) by (2.3). Then (a) and (b) holds for a Zariski dense subset of all operators g (i.e., for the integral isometries). Hence it holds for all g. Then by the uniqueness statement one observes (c). Again (d) follows by the Zariski density argument from the integral case (which is [13, Theorem 1.2(6)]). We also refer to [2, Proposition 5.1] for more details on extending the Markman operator from integral isometries to isometries defined over more general coefficient rings.
One can reinterpret the condition (f ⊗ g)(u v 1 ) = u v 2 in terms of generators of the cohomology ring. Following We write B k (x) for its component in degree 2k. In particular, Q) be a degree-preserving isometric ring isomorphism. Then the following are equivalent: Proof . Since g is an isometry of the Mukai lattice we have for x ∈ H * (S 1 ) the following equality in H * (M 2 ): Indeed, if we write u v 2 = i a i ⊗ b i under the Künneth decomposition, then Hence we see that: ⇐⇒ (a).

Universality
We apply Theorem 2.4 to study descendent integrals over M . Let k ≥ 0 and let P (t ij , u r ) be a polynomial depending on the variables t j,i , j = 1, . . . , k, i ≥ 1, and u r , r ≥ 1.
Let also A = (a ij ) k i,j=0 be a (k + 1) × (k + 1)-matrix. Our main result is the following.
In other words, the integral depends upon the above data only through P , the dimension dim M = 2n, and the pairings v · x i and x i · x j for all i, j. The proof of Theorem 2.9 will proceed in several steps. We begin with a general vanishing result. for some s i ∈ Z vanishes unless = 0.
Proof . We give two proofs of this fact. For the first proof, choose an isometry g : Λ C → Λ C such that where v · v = 2n − 2. Such an isometry exists since v · v > 0 and SO(Λ C ) acts transitively on vectors of the same square. By Theorem 2.4(a) for the first and Corollary 2.8 and Theorem 2.4(d) for the second equation, we find that By [1, Theorem 4.1] (or more precisely, the induction method used in the proof), this last integral depends upon w only through its intersection numbers against products of Chern classes of S and degree-components of gx j . 5 Since these intersections numbers are all zero, we may replace w by 0, in which case the claimed vanishing follows immediately.
Alternative proof . If w = 0 there is nothing to prove, so let w = 0. Choose w ∈ Λ C such that w · w = 1 and w · w = w · v = 0. Extend v, w, w to a basis {v, w, w } ∪ {e i } 24 i=4 of Λ C . For any j, expand x j in this basis: x j = a 1 v + a 2 w + a 3 w + a 4 e 4 + · · · + a 24 e 24 .
Because x j · w = 0, we must have a 3 = 0. By an induction on the number of classes x j , we know the claim of Proposition 2.10 if x j is a multiple of w. 6 Moreover, if we know the claim for x j ∈ {u 1 , u 2 } for some u 1 , u 2 ∈ Λ C then we know it for x j = u 1 + u 2 by expanding the monomial in (2.5). Hence we may replace x j by x j − a 2 w. In other words, we may assume that a 2 = 0. Doing so for all j, we hence see that w ∈ Λ C satisfies w · w = 1, w ⊥ Span(w , v, x 1 , . . . , x k ).
Consider the Lie algebra g = so v ⊥ ∼ = ∧ 2 v ⊥ . Theorem 2.4 induces a Lie algebra action γ : g → End H * (M ). By Theorem 2.4(a) γ(g) acts by derivations on H * (M ) and acts trivially on H 4n (M ). (This Lie algebra action is part of the Looijenga-Lunts-Verbistky Lie algebra action, see [13,Lemma 4.13].) Take w ∧ w ∈ g. Since the Lie algebra acts trivial on H 4n (M ) we have On the other hand, by Corollary 2.8 we have γ(w ∧w )B s i (w) = B s i (w) and γ(w ∧w )B i (x j ) = 0, and by Theorem 2.4(d) we have γ(w ∧ w )c r (T M ) = 0. Since γ(w ∧ w ) acts by derivations, we also get Lemma 2.11. In the situation of Theorem 2.9, there exists y i ∈ Λ C which have the same intersection matrix as in (2.4), satisfy and such that the span L = Span(v, y 1 , . . . , y k ) ⊂ Λ C is non-degenerate (i.e., the restriction of the inner product of Λ C onto L is non-degenerate).
Proof . Let L = Span(v, x 1 , . . . , x k ). Assume that L is degenerate, i.e., there exists a non-zero w ∈ L such that w · x i = 0 for all i and w · v = 0. Since v · v ≥ 2, we have that v, w are linearly independent. Hence they can be extended to a basis u 0 , . . . , u d of L with u 0 = w and u 1 = v. For every i let λ i ∈ C be the unique scalar such that We hence obtain Set y j = x j − λ j w. If Span(v, y 1 , . . . , y k ) is non-degenerate, we are done, otherwise repeat the above process. This process has to stop, since the dimension of the span drops by one in each step.
Proof . Let w 1 , . . . , w ∈ V be a list of vectors such that h ij = w i , w j is invertible. Pairing any linear relation between the w i 's with w j for j = 1, . . . , , and multiplying this system of equations by the inverse of h shows that the w 1 , . . . , w are linearly independent. This proves the first claim. For the second claim, we can choose a subset {w 1 , . . . , w d } ⊂ {v 1 , . . . , v k } which forms a basis of L = Span(v 1 , . . . , v k ) and observe that the matrix of the isomorphism L → L ∨ induced by the inner product with respect to the basis {w i } and the dual basis {w * i } is the Gram matrix of the w i . This shows that rank(g) ≥ dim L. Lemma 2.13. Let V be a finite-dimensional C-vectorspace with a C-linear inner product. Let v 1 , . . . , v k ∈ V and w 1 , . . . , w k ∈ V be lists of vectors such that (i) L = Span(v 1 , . . . , v k ) is non-degenerate, Then there exists an isometry ϕ : V → V such that ϕ(v i ) = w i for all i.
Proof . By Lemma 2.12 and assumptions (i) and (ii) we know that Choose a basis of L from the v 1 , . . . , v k , which we can assume is of the form v 1 , . . . , v d , where d = dim(L). By assumption (i) and Lemma 2.12 the gram matrix G : and similarly for any w ∈ M . The claim hence follows by writing every v i in this form, applying ϕ and using assumption (iii).

Case of dimension 2
We discuss how to evaluate integrals M τ k 1 (γ 1 ) · · · τ k (γ )P (c r (T M )), (2.6) whenever M = M H (v) is a 2-dimensional moduli space of stable sheaves, and hence a K3 surface. The universal family 7 F in this case induces a derived auto-equivalence The induced action on cohomology defines an isometry of Mukai lattices (in fact, a Hodge isometry), see [6,Chapter 16] for references for these well-known facts.

Proof of Theorem 1.1
If dim M > 2, the claim follows by Theorem 2.9 since (a) any descendent τ k (γ) defined as in (1.1) can been written as a polynomial in classes B j (x), and (b) for any list of vectors v, x 1 , . . . , x k ∈ Λ C after an isometry of Λ C we may assume that v is the Mukai vector which defines the Hilbert scheme of n points on a K3 surface. 9 If dim M = 2 and rk(v) > 0, as discussed in Section 2.4 any descendent τ k i (γ) can be written in terms of polynomials in classes Φ(α), where α is effectively determined by γ. Since any integral (2.6) can involve at most two classes of positive degree, this integral can be written as linear combination of the Mukai pairing between classes Φ(α) and Φ(α ) for various α, α . Since Φ is a Hodge isometry, these are just the Mukai pairings between α and α . This effectively determines the integrals (1.2). We also refer to Section 3.1 for a concrete implementation of this algorithm.
The case where dim(M ) = 2 and rk(v) = 0 is similar to the dim M = 2, rk(v) > 0 case, and left to the reader.

The Göttsche-Kool conjecture
Let S be a K3 surface and let M be a proper fine 2n-dimensional moduli space of stable sheaves on S of Mukai vector v. Let F be a universal family. We assume that rk(v) > 0. Our goal is to show that for any α ∈ K(S), class L ∈ H 2 (S) and u ∈ C we have where β ∈ K(S) is as specified in Theorem 1.2.
In Section 3.1, we first tackle the case dim M = 2 separately, and then afterwards prove the dim M > 2 case. Explicitly, the interesting pairings for the first three classes are The interesting intersections involving L are The pairings with up are u rk(v) times the pairings with p/ rk(v).