Lehn's conjecture for Segre classes on Hilbert schemes of points of surfaces and generalizations Lecture 1 : Hilbert schemes of surfaces in families. Rahul Pandharipande (1) Let p:S-> M be a flat family of nonsingular projective surfaces over any base M. (a) Examples to keep in mind * M is a point and S is a single surface, * p: S -> M is a CP2 bundle over M, * p: S -> M is the universal family of quasi-polarized K3 surfaces with quasi-polarization L. (b) Let h: S[n] -> M be the relative Hilbert scheme of n points in the fibers of p. The morphism h is smooth of relative dimension 2n. Let L -> S be a line bundle on the family of surfaces p: S -> M. For example, the K3 family of (a) above has a canonical choice L (up to normalization). (c) Given p: S -> M and L -> S , we can construct the tautological rank n vector bundle L[n] -> S[n] as usual with fiber H^0(Z,L) over the length n subscheme Z of of a fiber of p. Let c(L[n]) be the total Chern class in A*(S[n]) and let Seg(L[n]) = 1 / c(L[n]). Our main question is the following: How can we compute h_*(Seg(L[n]) in A^*(M)? Of course, we can also start with a vector bundle B -> S and ask to compute h_*(Seg(B[n]) in A^*(M). The most natural formulation of the problem is to start with an element of E of K^*(S) and ask to compute h_*(Seg(E[n]) in A^*(M). The K-theoretic formulation captures also h_*(c(E[n]) in A^*(M) since c(E)= Seg(-E[n]). (2) Why do we want to compute h_*(Seg(E[n]) ? (a) When M is a point, the answer is an integral over S[n] \int_{S[n]} Seg(E[n]) and has been studied via the Heisenberg algebra (Nakajima's theory of the cohomology of S[n]) by M. Lehn in 1999. He discovered there a nontrivial conjectural formula for the answer when E=L is a line bundle which has been now proven in the past few years by [MOP] for K3 surfaces and [Voisin] in general. * In case S is a K3 surface, we have complete results for all E in K^*(S) by [MOP]. * In case S is arbitrary and E in K^*(S) is of rank -2,-1,0,1, we also have complete results (needs [MOP], A. Mellit, [Voisin]). Rank 2 is almost complete. The geometric inputs have come from 3 very different directions * The virtual class of Quot scheme on S by [MOP], * Ideas of Reider/Lazarsfeld applied by Voisin, * Localization partition sums studied by Mellit. For general E the question, even for M is a point, is open. (b) The first result on these questions after Lehn's paper of 1999 was [MOP] in 2015. There the question of calculating h_*(Seg(L[n]) in A^*(M) for the universal family p: S -> M of quasi-polarized K3 surfaces arose naturally in the study of tautological relations on the moduli of K3 surfaces. To explain the relevance, we first ask: what are tautological classes on M? Here, we start with any family p: S -> M and and a line bundle L -> S. Let kappa[a,b,c] = p_*( c1(S)^a * c2(S)^b * c1(L)^c ) in A^*(M). Here c1(S) and c2(S) denote the Chern classes of the relative tangent bundle of the family p: S -> M. A tautological class on the base M is a polynomial in the classes { kappa[a,b,c] }_{all a,b,c >= 0} . Theorem 1: (Formulation already implicit in [MOP]) h_*(L[n]) in A^*(M) is given by universal polynomials in the classes { kappa[a,b,c] }. The Quot scheme method of [MOP] required the calculation of h_*(L[n]) in A^*(M) for the moduli space of K3 surfaces as a step in writing relations among the kappa[a,b,c] classes on M. Comments on Theorem 1: (i) The result shows h_*(E[n]) is a tautological class. (ii) Universal polynomial here means that the polynomial does not depend upon p: S -> M or E -> S Theorem 1 stated for any E in K^*(S) takes the form: h_*(E[n]) in A^*(M) is given by universal polynomials in the classes { kappa[a,b,C] } depending only on the rank of E. Here C is vector indexed by all the Chern classes of E. (iii) While a proof has not (yet) been written, the method is standard. Use the geometric arguments of Lehn's 1999 paper (studied there when M is a point) and apply them to the whole family h: S[n] -> M. (c) A parallel exists in the more developed study of tautological relations in the moduli space of curves. Let p: C -> M_g be the universal curve over the moduli space of nonsingular genus g curves. Let h: C[n] -> M_g be the universal Hilbert scheme of points. Push-forward calculations of, for example, h_*(w_C[n]) in A^*(M_g) play an import role in the study of tautological relations. The h_* calculation for C[n] is much easier than for S[n]. A fourth example (in addition to the three of part (1.a) for a family of surfaces is constructed from curves. Let C_1 -> M_g1 and C_2 -> M_g2 the universal families over the moduli spaces of nonsingular curves of genus g1 and g2. Let p: S = C_1 x C_2 -> M_g1 x M_g2 = M . Then Theorem 1 says, for example, that h_*( E[n] ) is tautological on M_g1 x M_g2, where E is obtained from the Hodge bundles and relative dualizing sheaves of the two moduli of curves. (3) The toric case (a) Theorem 1 tells us where the answer to our main question lives. But how can we calculate? Answer: Use toric geometry. The simplest case is to consider the surface S=C^2 the complex plane. Of course S is not compact. To compensate for the non-compactness, we must study the geometry equivariantly with respect to a torus action. Let the toruc T = C^* x C^* act on C^2 with tangent weights s,t at the origin. Let T also act on the trivial line bundle L = O_{C^2} -> C^2 equivariantly with weight w. A localization vertex for the Segre class push-forward is the following: Z(q,s,t,w) = \sum_{n=0}^{infty} q^n \int_{S[n]} Seg(L[n]). This generating series is defined by Atiyah-Bott localization with respect to T and is entirely explicit. When T acts on C^2[n], the T-fixed points are given by monomial ideals which are viewed as partitions in the plane. So we write Z(q,s,t,w) = \sum_{Sigma} q^{|Sigma|} Res_{Sigma} Seg(L[n]). where the sum is over all partitions Sigma (including the empty partition which contributes a leading 1 to the sum). The localization residue is 1 Res_{Sigma} Seg(L[n] = ------------------------------ e(Tan_Sigma) * c(O[n] \tensor w) where Tan_Sigma is the standard T-representation on the tangent space of C^2[n] at the T-fixed point Sigma (given by arm and leg lengths) and O[n] is the T-representation on the structure sheaf of the associated scheme. (b) Since Z starts with 1, we can take the log, F = Log(Z). Theorem 2. The logarithm F has the following basic structure F_0 F_1 F_2 F = ----- + ---- + ---- + ... st st st where F_0(q) is just a series in q, F_1(q,s,t,w) is of degree 1 in s,t,w, F_2(q,s,t,w) is of degree 2 in s,t,w, ... and the F_i are of course symmetric in s,t. So F_0(q) is really just 1 q-function, F_1(q) is 2 q-functions (coeffs of s+t and w), F_2(q) is 4 functions (coeffs of (s+t)^2, st, (s+t)*w, w^2), etc. There are 2 proofs of Theorem 2 * Mellit has a proof which directly depends upon analysis of the localization sum defining Z. * There is a second proof by geometry of log Hilbert scheme which I will explain. Finally, I note that Z here is not unrelated to the Nekrasov partition function --- the new aspect here is the inclusion of the Segre class. (c) How does Theorem 2 provide a calculation of h_*(Seg(L[n]) in A^*M for all families p: S -> M and line bundles L -> S ? The answer is given in a few steps. (i) Take the expansion of F = log(Z) in Theorem 2 and throw away F_0 and F_1: F_2 F_3 F_4 F# = ----- + ----- + ------ + ... . st st st (ii) Rewrite each term with kappa classes. Start with F_2 which is * (s+t)^2/st + * st/st + * (s+t)w/st + * w^2/st where the 4 instances of * denote 4 series in q (all without constant term). Write this as G2 = * kappa[2,0,0] + * kappa[0,1,0] + * kappa[1,0,1] + * kappa[0,0,2] where the q-series are the same as for F2. Similarly, write G3, G4, ... starting with F3, F4, ... by (s+t) => c1(S) st => c2(S) w => c1(L). (iii) Let G = G2 + G3 + G4 + ... where the right hand side is now a polynomial in the kappa classes { kappa[a,b,c] }_{a,b,c >=0} . Actually, only the kappa classes of codim >= 0 in M appear (since we have thrown out the negative dimensional kappas). (iv) Let p: S -> M be a family of nonsingular projective surfaces, and let L -> S be a line bundle. Theorem 3. We have the equality Exp(G) = \sum_{n=0}^{infty} h_*(Seg(L[n]) . Comments. The proof of Theorem 3 uses * Theorem 1 --- so we need only to find the universal formula. * Theorem 2 --- which gives the specialization to the projective toric case and a bit more. The more is because projective toric surfaces, unfortunately do not fully explore the tautological classes. To see this, consider the following exercise. Let S be a projective toric surface with T-action. Prove the following formula \int_S c1*c2 = 0 in T-equivariant cohomology. As a result, kappa[1,1,0] will always be zero in the projective toric case (and its contribution will be lost). A nice way to repair the defect in projective toric geometry is to consider toric surface relative to toric divisor (log toric situation). Then the issue can be overcome. (v) The localization strategy, including Theorems 2 and 3, are valid for K-theory classes of all ranks (not just the line bundle case discussed above). The statements and methods are the same. (d) The initial F_0 term has a very nice formula. Theorem 4 (Mellit) Consider the rank = r case, so Z(q,s,t,w_1,...,w_r) = 1 \sum_{Sigma} q^{|Sigma|} -------------------------------------------------------- e(Tan_Sigma) * c(O[n]\tensor w_1 + ... + O[n]\tensor w_r) Then the leading F_0/st term of F = log Z is F_0 = q + \sum_{n>=2} q^n (-1)^(n+1)*r*(r+1)^(n-1)*Binom((r+2)n-3,n-2)/(n^2*(n-1)). Mellit also has results on F_1. Both F_0 and F_1 are terms corresponding to "negative degree" kappa classes and do not play a direct role in the original question (they are removed in step (c.1) above). (e) Consider the rank 0 case (which is not at all trivial). But a special case of rank 0, when E is actually 0, is easy to understand. Then the parition function is 1 Z(q,s,t) = \sum_{Sigma} q^{|Sigma|} ------------ . e(Tan_Sigma) Recall the tangent representation Tan_Sigma is \sum_{Boxes b of Sigma} (Leg(b)+1)*s -Arm(b)*t) + \sum_{Boxes b of Sigma} -Leg(b)*s + (Arm(b)+1)*t . While the sum defining Z looks complicated, the answer is simple Z(q,s,t) = Exp(q/st) . Proof (Using the GW/DT correspondence of MNOP). Consider C^2 x P1. Then the DT theory of ideal sheaves in curve class d[P1] starts with Z_{C2xP1,DT} = q^d * Coeff(Z,q^d) + ... since the leading term corresponds to the Hilbert scheme of d points of C^2. The GW/DT correspondence here takes the form (-q)^d * Z_DT = (-iu)^{2d} * Z_GW , where Z_{C^2xP1,GW} is the Gromov-Witten partition function of C^2 x P1 is curve class d[P1]. For simple dimension reasons, the connected domain components of maps which can contribute to Z_GW are of genus 0 and map with degree 1 to P1. Hence, Z_GW = u^{-2d}/d! is the entire partition function. Using the GW/DT correspondence, (-iu)^{2d} * Z_GW = (-1)^d * 1/d! * (st)^{-d} = (-1)^d Coeff(Z,q^d) + higher order . We conclude (i) Coeff(Z,q^d) = 1/d! * (st)^{-d} (ii) There are no higher order terms. The claim Z = Exp(q/st) follows from (i). Conclusion (ii) is an interesting extra property. Bibliography [MOP] stands for arXiv: arXiv:1507.00688, arXiv:1708.0808129, arXiv:1712.02382 [Voisin] is arXiv:1708.06325