Corollary 9.8.26EB72DC4EC68

Proof. Let $f\colon Y\to X$ be finite and bimeromorphic, with $X$ normal, and let $A\subset X$ be a nowhere dense analytic subset such that $f$ is an isomorphism over $X\setminus A$ and $f^{-1}(A)$ is nowhere dense. Normal spaces are unibranch, so the finite fibers of $f$ are connected by theorem 9.8.1, hence consist of at most one point. The image of $f$ is closed, because $f$ is proper, and contains the dense set $X\setminus A$; so $f$ is a continuous closed bijection, that is, a homeomorphism. Let $x\in X$ and $y=f^{-1}(x)$. Choose a neighborhood $V$ of $y$, relatively compact in the domain of a local model of $Y$, so that the coordinate functions of the model are bounded on $V$. Composed with $f^{-1}$, they are bounded holomorphic functions on $f(V)\setminus A$, where $f^{-1}$ is holomorphic, and $f(V)$ is an open neighborhood of $x$. By lemma 9.4.4 they extend holomorphically to $f(V)$, and the extensions are the continuous functions $(\text{coordinates})\circ f^{-1}$. Hence $f^{-1}$ is holomorphic.

For the second assertion, over the dense open set $S\setminus \phi ^{-1}(X_{\mathrm{sing}})$ there is a unique lift $\psi _0=\nu ^{-1}\circ \phi $, because $\nu $ is an isomorphism over $X_{\mathrm{reg}}$. We extend it by the argument used for the uniqueness in the proof of theorem 9.4.6. Let $s\in S$ and $x=\phi (s)$, choose disjoint neighborhoods $W_ l$ of the finitely many points of $\nu ^{-1}(x)$, relatively compact in domains $W'_ l$ of local models of $\widetilde X$, and a neighborhood $U$ of $x$ with $\nu ^{-1}(U)\subset \bigcup _ lW_ l$, which exists by properness. Since $S$ is normal, hence unibranch, corollary 9.4.9 gives a neighborhood $V$ of $s$ with $\phi (V)\subset U$ such that $V\setminus \phi ^{-1}(X_{\mathrm{sing}})$ is connected. Then $\psi _0$ maps this set into a single $W_ l$. The coordinate functions of $W'_ l$ composed with $\psi _0$ are bounded and extend holomorphically to $V$, by lemma 9.4.4; the extended map takes values in $\overline{W_ l}\subset W'_ l$, and it satisfies the equations of the local model, which hold on a dense subset of $V$. This gives a holomorphic lift on $V$, with $\nu \circ \psi =\phi $ by density. Lifts agree on the dense set where $\psi _0$ is defined, so these local lifts glue, and the lift is unique. $\square$