Refined Verlinde formulas for Hilbert schemes of points and moduli spaces of sheaves on K3 surfaces

We compute generating functions for elliptic genera with values in line bundles on Hilbert schemes of points on surfaces. As an application we also compute generating functions for elliptic genera with values in determinant line bundles on moduli spaces of sheaves on K3 surfaces.


Introduction
The celebrated Verlinde formula (see [Ver88,NR93,BL94,Fal94]) is a formula for the generating function for dimensions of spaces of sections of line bundles on moduli spaces of vector bundles on algebraic curves. Now let S be a smooth projective surface over C and S [n] the Hilbert scheme of n points on S. For every vector bundle V on S there is a corresponding tautological bundle V [n] of rank rk(V [n] ) = n rk(V ), whose fibre over Z ∈ S [n] is H 0 (Z, V | Z ). The map V → V [n] extends to a homomorphism from the Grothendieck group K 0 (S) of vector bundles on S to K 0 (S [n] ). For L ∈ S [n] denote µ(L) := det((L − O S ) [n] ) and E := det(O [n] S ). Then it is well known that Pic(S [n] ) = µ(Pic(S)) ⊕ ZE. The analogue of the Verlinde formula for Hilbert schemes of points is a formula for the generating function for holomorphic Euler characteristics χ(S [n] , µ(L) ⊗ E r ). In [EGL01] such a formula was proven in the cases r = −1, 0, 1 or K 2 S = K S L = 0. On the other hand the celebrated Dijkgraaf-Moore-Verlinde-Verlinde formula [DMVV97], shown in [BL00, BL03,BL05], relates the generating function of the elliptic genera Ell(S [n] ) of Hilbert schemes of points to Siegel modular forms.
In this short note we interpolate between these two results, by proving a formula for Ell(S [n] , µ(L) ⊗ E r ), the elliptic genus with values in the line bundle µ(L) ⊗ E r . To state these results we introduce the following power series. φ −2,1 (q, y) := (y 1/2 − y −1/2 ) 2
Then the DMVV formula says The first theorem deals with the case r = 0.
Theorem 1.1. Let S be an algebraic surface, L ∈ Pic(S). Then Specializing to L = O S , we recover the DMVV formula. Specializing to q = 0 yields an infinite product formula for the χ y -genera with values in a µ(L), which in turn recovers for L = O S the formula for the χ −y -genera of Hilbert schemes from [GS93]. We write For general line bundles on Hilbert schemes of points we can partially determine the generating function.
Specialising to q = 0 again yields a formula for the χ −y -genus with values in µ(L) ⊗ E r . (1 − z n ) 2 (1 − z n y)(1 − z n /y) n 2 2 (L 2 +r 2 χ(O S )) with the change of variables p = z n>0 (1−z n ) 2 (1−z n y)(1−z n /y) Finally we show an analogue of Theorem 1.3 for moduli spaces of sheaves on K3 surfaces S. Fix s ∈ Z >0 . Let H be an ample line bundle on S and M H S (s, c 1 , c 2 ) the moduli space of H-semistable sheaves on S of rank s with Chern classes c 1 , c 2 . We assume that M H S (s, c 1 , c 2 ) consists only of stable sheaves. For any choice of s, c 1 , c 2 , we denote vd = vd(s, c 1 , c 2 ) := 2sc 2 − (s − 1)c 2 1 − 2(s 2 − 1).
). This is the generalization of the line bundle with the same name on S [n] = M H S (1, 0, n). We obtain the following result.
Theorem 1.5. Let S be a K3 surface, s ∈ Z >0 . Under the assumptions above we have with the change of variables p = zL (2,0) (−φ −2,1 , z) r 2 s 2 , and with the change of variables p = z n>0 In the special case of K3 surfaces Theorem 1.5 in particular confirms the conjectures of [GKW] about refinements of Verlinde formulas for moduli spaces of rank 2 sheaves on surfaces in the case of K3 surfaces. The specialization L = O X reproduces in the case of K3 surfaces the formulas of [GK] on the elliptic genus of moduli spaces of sheaves on surfaces.
Acknowledgements. I thank Don Zagier for helping me with the proof of Lemma 4.3. This work grew out of collaboration with Martijn Kool. I thank him for many useful discussions.

Hilbert schemes of points
Let S be a smooth projective surface. We denote S [n] the Hilbert scheme of n points on S. It is a smooth projective variety of dimension 2n. Let S (n) be the n-th symmetric power of S. The Hilbert-Chow morphism π : S [n] → S (n) , Z → supp(Z), sending a zero dimensional scheme to its support with multiplicities is a crepant resolution of S (n) , i.e. it is birational and π * K S (n) = K S [n] . Let Z n (S) ⊂ S × S [n] be the universal subscheme, with projections p : Z n (S) → S [n] , q : Z n (S) → S. For a vector bundle V of rank r on S the corresponding tautological vector bundle is V [n] := p * q * V , a vector bundle of rank rn on S [n] . This extends to a homomorphism [n] : K 0 (S) → K 0 (S [n] ) between the Grothendieck groups of vector bundles. We put E := det(O ). Let η : S n → S (n) be the natural projection. Let L n := η * (⊗ n i=1 pr * i L) S n be the S n -equivariant pushforward, where pr i : S n → S is the i-th projection. Then it is well-known that µ(L) = η * (L n ), and from the definitions it follows that det(V [n] ) = µ(det(V )) ⊗ E rk(V ) .

Elliptic genus.
For a compact complex manifold M, the χ −y -genus is Usually we consider the normalized version For a rank r vector bundle V on M put Write y = e 2πiz , q = e 2πiτ . Then for W ∈ K 0 (M), the elliptic genus and the elliptic genus with values in W are defined by be the classical Jacobi theta function, where we write y = e 2πiz , q = e 2πiτ . Let c(M) = n j=1 (1 + x j ) be a formal splitting of the total Chern class of M. Putting it follows from the definitions and the Riemann-Roch theorem that

Beauville-Bogomolov quadratic form
Let X be a compact holomorphic symplectic manifold of dimension dim(X) = 2m. We briefly review some properties of the Beauville-Bogomolov quadratic form q X : H 2 (X) → Q on X from [Huy99, Sections 1.9-1.11]. Note that the odd Chern classes of X vanish.
Theorem 2.1. For any β ∈ H 4k (X, Q) in the sub-algebra generated by the Chern classes of X, there is a constant c(β) ∈ Q, such that for all α ∈ H 2 (X, Q) The quantity c(β) is invariant under deformation of X.

Corollary 2.2. There exists a polynomial h X
The polynomial h is invariant under deformation of X.
In this section we will prove Theorem 1.1. We start by reviewing some of the ideas and definitions of [BL05]. If X is a nonsingular projective variety, with an action of a finite group their formula specializes to Here Z runs over the irreducible components of the common fixpoint set of g and h, and [Z] is the class of Z in the Chow group of X. The restriction of T X to Z splits into linearized bundles according to the Now consider the action of S n on S n , and recall the quotient morphism η : S n → S (n) and the Hilbert Chow morphism π : S [n] → S (n) . As S [n] is nonsingular we have ELL(S [n] ) = Ell orb (S [n] ). As π is a crepant resolution, [BL05, Theorem 3.5] implies that π * ELL(S [n] ) = Ell orb (S (n) ). Thus we find by the above In the second line we have used π * (L n ) = µ(L) and the projection formula, and in the third line we have used (3.2), η * (L n ) = ⊗ n i=1 pr * i L and again the projection formula.
Let x 1 , x 2 be the Chern roots of T S . Then Now we prove Theorem 1.1 by adapting the proof of [BL05, Theorem 6.1]. Note that in the notations of [BL05] we have D = 0, which leads to many simplifications.
Let (g, h) be a commuting pair in S n . We sum up the description of the action of g, h and their fixpoint sets in the proof of [BL05, Theorem 6.1]. We have a decomposition {1, . . . , n} = J 1 ∪ . . . ∪ J m into the orbits of the subgroup generated by g, h. Thus the action of (g, h) on S n restricts to an action on each of the corresponding products S J l . Furthermore we can write |J l | = a l b l for positive integers a l , b l , and up to reordering of the elements of J l the action of (h, g) on S J l can be described as follows. Write (y i,j ) i∈Z/a l Z, j∈Z/b l Z the components of elements of y ∈ S J l . Then the action of g, h on S J l is given by h(y i,j ) = y i,j+1 , g(y i,j ) = y i+1,j for 0 ≤ i < a l − 1, g(y a l −1,j ) = y 0,j+s , for some s ∈ {0, . . . , b l − 1}, and s determines the action of (g, h) on J l uniquely. The fixpoint set (S J l ) g,h is S embedded via the diagonal map j l : S → S J l .
Changing their notation slightly, we denote for (a, b, s) := (a l , b l , s l ) by .
Note that the left hand side is where ( ) k denotes the part in degree k. As j * l η * (L n ) = L ab , we obtain By definition it is clear that the contribution of (g, h) to S n Ell orb (S n , S n )η * L n is This proves Theorem 1.1.

The case of Hilbert schemes of points on K3 surfaces
Now we want to consider the case of Hilbert schemes of points on K3 surfaces. We obtain a formula for all r. It is shown in [Bea83,lem. 9.1] that for L ∈ Pic(S) we have (4.2) q S [n] (L + rE) = L 2 − 2r 2 (n − 1).
where ELL is the genus associated to a power series, and µ(L) ⊗ E r = det((L + (r − 1)O S )) [n] . Therefore [EGL01, Theorem 4.2] applies and gives the following.
Thus Theorem 1.3 follows from the following lemma. where p = zg(z) k .
Proof. Without loss of generality we can assume that w = 0 (otherwise replace f (p) by g(p) w f (p)), and k = 1 (otherwise replace g(p) by g(p) k and note that z d dz log(g(z) k ) = kz d dz log(g(p))). We can describe h(p) as follows: for a variable u write g(p) −u f (p) = n≥0 h n (u)p n , with h n (u) a polynomial in u, then h(p) = n≥0 h n (p d dp )p n , i.e. move all factors of u to the left and then replace u by p d dp . We make the variable transformation p = e x , so that p p dp = d dx , and we write g(p) = e a(x) , f (p) = φ(x). Then we obtain .
In the second line we have used the Lagrange inversion formula In the third line we put z := e η , thus p = e x = e η+a(η) = zg(z).

Moduli of sheaves on K3 surfaces
In this section we extend our results to moduli spaces of sheaves on K3 surfaces. First we briefly recall determinant line bundles on moduli spaces of sheaves, for details see e.g. [GNY09, Section 1.1], [HL10,Chapter 8].
For a Noetherian scheme Y denote by K(Y ) and K 0 (Y ) the Grothendieck groups of coherent sheaves and locally free sheaves. If Y is nonsingular and projective, then K(Y ) = K 0 (Y ). We denote by [F ] the class of a sheaf F in K(Y ). For a proper morphism f : Z → Y the pushforward f ! : Now let S be a smooth projective surface. On K(S) there is a quadratic form (u, v) → χ(S, u ⊗ v) to be denoted by χ(u ⊗ v). Classes u, v ∈ K(S) are called numerically equivalent if u − v is in the radical of this quadratic form. Let K(S) num be the set of numerical equivalence classes. Let c ∈ K(S) num . For a flat family E of coherent sheaves on S of class c ∈ K(S) num parametrized by a scheme T , let q : S × T → S, p : S × T → T be the projections and define λ E : K(S) → Pic(S) by the composition Let φ : K(S) → H * (S, Z) be the homomorphism defined by φ(E) = ch(E) td(S). Let w := φ(c), then φ induces a injective homomorphism φ : K c → w ⊥ . There is a homomorphism θ w : w ⊥ → H 2 (M H S (c), Z), such that θ w (φ(v)) = c 1 (λ(v)) for all v ∈ K c . We have assumed that rk(c) > 0 or that rk(c) = 0 and c 1 (c) is nef and big, furthermore vd(c) > 1, and M H S (c) consists only of stable sheaves. Under these assumptions we know ([GNY09, Theorem 1.14], [Yos01a], [Yos01b]), that M H S (c) is an irreducible symplectic manifold which is deformation equivalent to S [vd(c)/2] , θ w : w ⊥ → H 2 (M H S (c), Z) is surjective, and for x in w ⊥ we have x, x = q M H S (c) (θ w (x)).