Virtual normalization and virtual fundamental classes

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.