# Virtual classes of $\mathbb{G}_\text{m}$-gerbes

## TL;DR

This paper establishes a relationship between the obstruction theories of $ ext{G}_m$-gerbes and their bases, simplifying the construction of virtual classes and advancing the understanding of their geometric properties.

## Contribution

It introduces a method to derive semi-perfect obstruction theories for the base of a $ ext{G}_m$-gerbe from its perfect obstruction theory, with conditions for perfection.

## Key findings

- A perfect obstruction theory for a $ ext{G}_m$-gerbe induces a semi-perfect obstruction theory for its base.
- The semi-perfect obstruction theory for the base is perfect if the gerbe is quasi-compact and affine-pointed.
- The results relate the virtual classes of the gerbe and its base.

## Abstract

We show that a perfect obstruction theory for a $\mathbb{G}_\text{m}$-gerbe determines a semi-perfect obstruction theory for its base, which is perfect if the gerbe is quasi-compact and affine-pointed. These results streamline the construction of a semi-perfect obstruction theory for the base, and allow us to relate virtual classes of the gerbe and its base.