Quasimaps to moduli spaces of sheaves

Abstract

We develop the theory of quasimaps to moduli spaces of stable sheaves on a surface. Under some natural assumptions we prove that the moduli spaces of quasimaps are proper and carry a perfect obstruction theory. Moreover, the moduli spaces of quasimaps are naturally isomorphic to certain relative moduli spaces of sheaves. Using the theory of entangled tails we establish the wall-crossing formulae which relate Donaldson-Thomas theory of threefolds of the type Surface×Curve to Gromov-Witten theory of moduli spaces of sheaves on the surface. Contents 1. Introduction 1 2. Stack of coherent sheaves 10 3. Quasimaps 12 4. Hilbert schemes 28 5. Obstruction theory 35 6. Wall-crossing 42 Appendix A. Stability of fibers 50 References 554 By which we mean 0 + -stable quasimaps.5 Those points that are mapped outside of the stable locus. 6 The degree is weighted more than the Euler characterstics.7 More precisely, the stack Coh(S) is locally a GIT stack. However, it is unbounded and

A Lie algebra action on the Chow ring of the Hilbert scheme of points of a K3 surface

“…In the last section we saw the relationship between the theories (i) and (ii). 5 In this section we review the wall-crossing formula between (ii) and (iii) obtained by Nesterov [42] using the theory of quasimaps. Our discussion follows the survey paper [44].…”

Holomorphic anomaly equations for the Hilbert scheme of points of a K3 surface

“…By Lemma 4.2 the series ( 58) is a Laurent polynomial in p. Proof. Denis Nesterov in [67,68] showed that the left hand side is equal to a partition function of relative Pandharipande-Thomas invariants of (S × C, S z ), see in particular [68,Cor.4.5]. The statement follows then from the GW/PT correspondence for (S × C, S z ) proven in [72, Thm.1.2] whenever β is primitive.…”

“…The cycle Z S [n] (p, q) also appears naturally in the Pandharipande-Thomas theory of the relative threefold (S × P 1 , S 0,∞ ). Indeed, by Nesterov's quasi-map wallcrossing [67,68] and the computation of the wall-crossing term in [81] one has…”

“…For K3 × P 1 it was recently established in [72] for curve classes which are primitive over the surface. The Hilb/PT correspondence (between (i) and (ii)) was recently established in full generality by Nesterov [67]. For C 2 and resolutions of A n singularities the triangle of correspondences was worked out previously in [84,17,85,58,55,59,50].…”

“…For the current work several new ingredients are required, which have not been available in [69]. Most notably are (without order): holomorphic anomaly equations for K3 surfaces [78,79], Nesterov's quasi-map wall-crossing [67,68], the double cosection argument of [90] leading to [81], new methods to deal with vanishing cohomology in Gromov-Witten and Pandharipande-Thomas theory [2,72], the description of the monodromy of S [n] in terms of Nakajima operators [71], and new ideas to prove modularity of Jacobi forms [76,80].…”