Dimensional reduction in cohomological Donaldson-Thomas theory

TL;DR

This paper proves a dimensional reduction theorem linking hypercohomology of certain perverse sheaves on shifted cotangent stacks to Borel-Moore homology, with applications to Donaldson-Thomas invariants and virtual fundamental classes.

Contribution

It establishes a global dimensional reduction theorem for perverse sheaves in cohomological Donaldson-Thomas theory, extending Davison's local result.

Findings

Hypercohomology is isomorphic to Borel-Moore homology up to a degree shift.

Application to cohomological Donaldson-Thomas invariants for local surfaces.

Proposes a sheaf-theoretic construction of virtual fundamental classes.

Abstract

For oriented $-1$-shifted symplectic derived Artin stacks, Ben-Bassat-Brav-Bussi-Joyce introduced certain perverse sheaves on them which can be regarded as sheaf theoretic categorifications of the Donaldson-Thomas invariants. In this paper, we prove that the hypercohomology of the above perverse sheaf on the $-1$-shifted cotangent stack over a quasi-smooth derived Artin stack is isomorphic to the Borel-Moore homology of the base stack up to a certain shift of degree. This is a global version of the dimensional reduction theorem due to Davison. We give two applications of our main theorem. Firstly, we apply it to the study of the cohomological Donaldson-Thomas invariants for local surfaces. Secondly, regarding our main theorem as a version of Thom isomorphism theorem for dual obstruction cones, we propose a sheaf theoretic construction of the virtual fundamental classes for quasi-smooth derived Artin stacks.