Virtual normalization and virtual fundamental classes

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

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