Virtual normalization and virtual fundamental classes

TL;DR

This paper compares two types of virtual fundamental classes in algebraic geometry, using virtual normalization and Costello's push-forward formula to establish their relationship.

Contribution

It introduces a method to identify virtual fundamental classes of different moduli stacks via virtual normalization and Costello's push-forward formula.

Findings

Virtual fundamental classes are identified via virtual normalization.

Costello's push-forward formula is applied to relate the classes.

The comparison enhances understanding of ramified and log stable maps.

Abstract

In this paper, we compare the virtual fundamental classes of the stacks of $(g,\beta,\mu)$-stable ramified maps $\bar{\mathfrak U}_{g,\mu}(X,\beta)$ and of $(g,\beta,\mu)$-log stable ramified maps $\bar{\mathfrak U}^{\mathrm{log}}_{g,\mu}(X,\beta)$. For that we will see how they are identified via virtual normalization and then apply Costello's push-forward formula.