Proposition 10.3.94C3B45D704A9

Proof. The manifold $Y=\pi ^{-1}(X)$ is compact, because $\pi $ is proper. By the definition of bimeromorphic maps (section 9.4), there is a nowhere dense analytic subset $E_0\subset X$ such that $\pi ^{-1}(E_0)$ is nowhere dense and $\pi $ restricts to an isomorphism of $Y\setminus \pi ^{-1}(E_0)$ onto $X\setminus E_0$. The set $X\setminus E_0$ is connected by lemma 9.2.9, hence so is its dense preimage, and therefore $Y$ is connected. The image $\pi (Y)$ is closed, since proper maps are closed, and it contains the dense set $X\setminus E_0$; so $\pi $ is surjective. By proposition 9.8.3 there is an analytic subset $A\subset X$ of codimension at least two such that $\pi $ is an isomorphism over $X\setminus A$. Then $\pi ^{-1}(A)$ is an analytic subset of $Y$ different from $Y$, because $\pi $ is surjective. Since $Y$ is connected, $\pi ^{-1}(A)$ is locally contained in the zero set of a holomorphic function that does not vanish identically on a connected neighborhood (section 9.2), so it is nowhere dense and a null set (lemma 5.5.1). The same holds for $A$ in $X$.

If $\varphi \in \mathrm{PSH}(X,\theta )$, let $B$ be a coordinate ball and $h$ a smooth function on $B$ with $\mathrm{dd}^{\mathrm{c}}h=\theta $ (lemma 6.14.2). On every connected coordinate neighborhood $W\subset \pi ^{-1}(B)$ the function $(\varphi +h)\circ \pi $ is plurisubharmonic or identically $-\infty $, by proposition 5.9.3, and the second case is excluded because $\varphi +h$ is finite almost everywhere on the image under the biholomorphism $\pi $ of the nonempty open set $W\setminus \pi ^{-1}(A)$. Hence $\varphi \circ \pi =(\varphi +h)\circ \pi -h\circ \pi $ is quasi-plurisubharmonic, with $\pi ^*\theta +\mathrm{dd}^{\mathrm{c}}(\varphi \circ \pi )=\mathrm{dd}^{\mathrm{c}}((\varphi +h)\circ \pi )\geq 0$ on $\pi ^{-1}(B)$. Thus $\varphi \circ \pi \in \mathrm{PSH}(Y,\pi ^*\theta )$. If $\varphi \circ \pi =\varphi '\circ \pi $, then $\varphi =\varphi '$ on $X\setminus A$, hence almost everywhere, and therefore everywhere, as noted at the beginning of this section; this proves injectivity.

Conversely, let $\psi \in \mathrm{PSH}(Y,\pi ^*\theta )$. Over $X\setminus A$ it descends to the function $\psi \circ \pi ^{-1}$, which is $\theta $-plurisubharmonic there. It is bounded above, because $\psi $ is upper semicontinuous on the compact manifold $Y$. On a coordinate ball $B$ with $\mathrm{dd}^{\mathrm{c}}h=\theta $ on $B$, the plurisubharmonic function $\psi \circ \pi ^{-1}+h$ on $B\setminus A$ is therefore locally bounded above near $A$, and it extends to a plurisubharmonic function on $B$ by theorem 5.11.1. The extensions on two balls agree outside the null set $A$, hence everywhere, so they define $\varphi \in \mathrm{PSH}(X,\theta )$. The pullback $\varphi \circ \pi $ equals $\psi $ outside the null set $\pi ^{-1}(A)$, hence everywhere. This proves surjectivity.

For continuity, we use proposition 10.3.1 on $X$ and on $Y$; by its last assertion, its assertions for a fixed background hold on $Y$. Let $\varphi _ j\to \varphi $ in $L^1(X)$. By the continuity of the supremum, it suffices to treat normalized functions, $\sup _ X\varphi _ j=\sup _ X\varphi =0$, since subtracting the suprema preserves $L^1$ convergence. The pullbacks have the same suprema, because $\pi $ is surjective, so they are relatively compact in $L^1(Y)$, and every cluster value lies in $\mathrm{PSH}(Y,\pi ^*\theta )$. On compact subsets of $Y\setminus \pi ^{-1}(A)$, a change of variables shows that every $L^1$ cluster value of $(\varphi _ j\circ \pi )$ coincides almost everywhere with $\varphi \circ \pi $; since $\pi ^{-1}(A)$ is a null set, every cluster value is $\varphi \circ \pi $. A relatively compact sequence in a metric space with a single cluster value converges to it; hence $\varphi _ j\circ \pi \to \varphi \circ \pi $ in $L^1(Y)$.

Finally, $\nu (\varphi \circ \pi ,E)$ is the infimum over $y\in E$ of the point Lelong numbers $\nu (\varphi \circ \pi ,y)$. Near $y$ choose a smooth $g$ with $\mathrm{dd}^{\mathrm{c}}g=\pi ^*\theta $, so that the functions $\varphi \circ \pi +g$ are plurisubharmonic. By (6.7.10), applied to them, each $\nu (\varphi \circ \pi ,y)$ is upper semicontinuous for $L^1$ convergence of $\varphi \circ \pi $, hence, by the continuity just proved, for $L^1$ convergence of $\varphi $; and an infimum of upper semicontinuous functions is upper semicontinuous. Since Lelong numbers do not change when a constant is added, we may restrict to the normalized functions, which form a compact set by proposition 10.3.1. An upper semicontinuous function with finite values attains its maximum on a compact set, which gives the uniform bound. $\square$