Localizing virtual fundamental cycles for semi-perfect obstruction theories

TL;DR

This paper extends localization techniques for virtual fundamental classes to semi-perfect obstruction theories, enabling broader applications such as the Jiang-Thomas virtual signed Euler characteristic without quasi-smoothness assumptions.

Contribution

It generalizes torus and cosection localization methods to semi-perfect obstruction theories, expanding their applicability in algebraic geometry.

Findings

Localization techniques are applicable to semi-perfect obstruction theories.

The Jiang-Thomas theory of virtual signed Euler characteristic is validated without quasi-smoothness.

The work broadens the scope of virtual fundamental class applications.

Abstract

Recently H.-L. Chang and J. Li generalized the theory of virtual fundamental class to the setting of semi-perfect obstruction theory. A semi-perfect obstruction theory requires only the local existence of a perfect obstruction theory with compatibility conditions. In this paper, we generalize the torus localization, the cosection localization and their combination, to the setting of semi-perfect obstruction theory. As an application, we show that the Jiang-Thomas theory of virtual signed Euler characteristic works without the technical quasi-smoothness assumption from derived algebraic geometry.