Quasimaps to moduli spaces of sheaves on a K3 surface

Abstract

We continue the study of quasimaps to moduli spaces of sheaves, concentrating this time on K3 surfaces. We construct a surjective cosection of the obstruction theory for sheaves on K3×Curve, using the semiregularity map. The novelty of our considerations lies in the fact that we consider non-commutative fist-order deformations of the surface to prove the surjectivity of the semiregularity map. We then proceed to proving the quasimap wall-crossing formulae for reduced classes. As applications we prove the wall-crossing part of Igusa cusp conjecture; higher-rank/rank-one DT wall-crossings on some threefolds of the type K3×Curve; relative DT/PT correspondence for K3×P 1 . Contents 1. Introduction 1 2. Surjective cosection 4 3. Wall-crossing 10 4. Applications 12 Appendix A. Reduced obstruction theory 18 References 22

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

“…In (68) the sum over r is finite by Lemma 4.2, hence the statement of the theorem is well-defined. If a < 0 in Theorem 10.1, then taut = 0, so all Gromov-Witten invariants would vanish.…”

“…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…”

“…In the case of K3 surfaces the above correspondences take the simplest form: they are straight equalities, without wallcrossing corrections [68]. By applying the product formula in the Gromov-Witten theory of S × P 1 we hence can express the invariants of the Hilbert scheme in terms of those of the K3 surface S. This allows us to lift modular properties known for K3 surfaces (by [63]) to the Hilbert scheme of points.…”

“…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].…”