Quasimaps to moduli spaces of sheaves on a K3 surface

DOI10.1017/FMS.2024.48arXiv2111.11425MaRDI QIDQ6383685

Publication date: 22 November 2021

Abstract: This article is the second in a series of three articles, the aim of which is to study various correspondences between four enumerative theories associated to a surface

S

: Gromov-Witten theory of

S^{[n]}

, orbifold Gromov-Witten theory of

[S^{(n)}]

, relative Gromov-Witten theory of

S i m e s C

for a nodal curve

C

and relative Donaldson-Thomas theory of

S i m e s C

. In this article, we focus on quasimaps to moduli spaces of sheaves on a K3 surface

S

. We construct a surjective cosection of the obstruction theory of moduli spaces of

e p s i l o n

-stable quasimaps. We then establish reduced wall-crossing formulas, which relate reduced Gromov-Witten theory of moduli spaces of sheaves on

S

and reduced Donaldson-Thomas theory of

S i m e s C

. As applications, we prove the wall-crossing part of Igusa cusp form conjecture; higher-rank/rank-one relative Donaldson-Thomas correspondence on

S i m e s C

, if

g (C) l e q 1

; relative Donaldson-Thomas/Pandharipande-Thomas correspondence on

S i m e s m a t h b b P^{1}

.

Mathematics Subject Classification ID

Vector bundles on surfaces and higher-dimensional varieties, and their moduli (14J60) (K3) surfaces and Enriques surfaces (14J28) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) (14N35) (3)-folds (14J30) Elliptic surfaces, elliptic or Calabi-Yau fibrations (14J27) Moduli, classification: analytic theory; relations with modular forms (14J15) Holomorphic symplectic varieties, hyper-Kähler varieties (14J42)