Virtual normalization and virtual fundamental classes

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