Virtual classes via vanishing cycles

TL;DR

This paper introduces a novel method for constructing virtual fundamental classes for quasi-smooth derived schemes using vanishing cycles, connecting to existing theories and extending to DT4 classes.

Contribution

It develops a new approach based on perverse sheaves and Fourier-Sato transform, unifying and extending previous virtual class constructions.

Findings

Classes coincide with Behrend-Fantechi and Li-Tian classes under certain conditions

Provides a new perspective using vanishing cycles and shifted cotangent spaces

Discusses potential for constructing DT4 virtual classes

Abstract

We develop a new method to construct the virtual fundamental classes for quasi-smooth derived schemes using the perverse sheaves of vanishing cycles on their $-1$-shifted contangent spaces. It is based on the author's previous work that can be regarded as a version of the Thom isomorphism for $-1$-shifted cotangent spaces. We use the Fourier-Sato transform to prove that our classes coincide with the virtual fundamental classes introduced by the work of Behrend-Fantechi and Li-Tian, under the quasi-projectivity assumption. We also discuss an approach to construct DT4 virtual classes for $-2$-shifted symplectic derived schemes using the perverse sheaves of vanishing cycles.