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 ${\bar{\mathfrak{U}}}_{g,\mu }(X,\beta )$ and of $(g,\beta ,\mu )$-log stable ramified maps $\bar{\mathfrak U}^{$\mathrm{l}\mathrm{o}\mathrm{g}$}_{g,\mu}(X,\beta)$. For that we will see how they are identified via virtual normalization and then apply Costello’s push-forward formula.