Abstract

Fundamental examples of symplectic varieties are moduli spaces of sheaves on K3 surfaces. This can be extended to higher-dimensional Calabi-Yau varieties through the concept of shifted symplectic structures in derived algebraic geometry. In the first half of the talk, I will introduce a general operation of producing shifted symplectic stacks from given ones. Basic examples like cotangent bundles, critical loci, and Hamiltonian reduction can be understood as special cases of this operation. Moreover, this unification enables us to provide an etale local structure theorem for shifted symplectic Artin stacks. In the second half of the talk, I will provide an application to enumerative geometry. I will explain the construction of Lagrangian classes for perverse sheaves in cohomological Donaldson-Thomas theory, whose existence was conjectured by Joyce. As examples, I will explain how to construct the following structures from the Lagrangian classes: (1) cohomological field theories for gauged linear sigma models; (2) cohomological Hall algebras for 3-Calabi-Yau categories; (3) relative Donaldson-Thomas invariants for Fano 4-folds with anti-canonical divisors; (4) refined surface counting invariants for Calabi-Yau 4-folds. This is joint work in progress with Adeel Khan, Tasuki Kinjo, and Pavel Safronov.