Proposition 6.12.458995E4B098A

Proof. By proposition 9.8.3, $\pi $ is biholomorphic over $X\setminus S$ for an analytic subset $S\subset X$ of codimension at least two; in particular $S$ is nowhere dense. The analytic subset $\pi ^{-1}(S)$ of $Y$ is nowhere dense as well, because every nonempty open subset of $Y$ contains a nonempty open set that $\pi $ maps biholomorphically onto an open subset of $X$, and such a set is not contained in $S$. Near each of their points, nowhere dense analytic subsets lie in the zero set of a holomorphic function that does not vanish identically on a connected neighborhood (section 9.2); so $S$ and $\pi ^{-1}(S)$ are Lebesgue null sets in charts, by lemma 5.5.1.

In holomorphic coordinates the real Jacobian determinant of $\pi $, that is, the factor by which $\pi ^*$ multiplies the volume form $\mathrm dx_1\wedge \mathrm dy_1\wedge \cdots \wedge \mathrm dx_ n\wedge \mathrm dy_ n$, is $|J_\pi |^2$. Indeed, since $\frac{\mathrm i}2\mathrm dz_ j\wedge \mathrm d\bar z_ j=\mathrm dx_ j\wedge \mathrm dy_ j$, this form equals $c_ n\, \mathrm dz_1\wedge \cdots \wedge \mathrm dz_ n\wedge \mathrm d\bar z_1\wedge \cdots \wedge \mathrm d\bar z_ n$ with $c_ n=(\mathrm i/2)^ n(-1)^{n(n-1)/2}$, the sign coming from moving each $\mathrm dz_ j$ to the left past $\mathrm d\bar z_1,\ldots ,\mathrm d\bar z_{j-1}$, and similarly in the coordinates $w$. Since $\pi ^*(\mathrm dz_1\wedge \cdots \wedge \mathrm dz_ n)=J_\pi \, \mathrm dw_1\wedge \cdots \wedge \mathrm dw_ n$ and the conjugate identity holds for the $\mathrm d\bar z_ j$, the claim follows. The change of variables formula for nonnegative measurable functions, applied to the biholomorphism $\pi $ from $W\setminus \pi ^{-1}(S)$ onto its image, therefore gives

for every nonnegative measurable $g$ and every open set $W$ in a coordinate chart of $Y$ that is mapped into a coordinate chart of $X$, with $J_\pi $ computed in these coordinates.

Now suppose that $|f|^2\mathrm e^{-\varphi }$ is integrable on a coordinate neighborhood $V$ of $x$. Applying the formula with $g=|f|^2\mathrm e^{-\varphi }$ to small coordinate neighborhoods $W\subset \pi ^{-1}(V)$, we see that (6.12.5) is integrable near every point of $\pi ^{-1}(V)$. Conversely, suppose that (6.12.5) is integrable near every point of $\pi ^{-1}(x)$, where $f$ is defined on a coordinate neighborhood $V_0$ of $x$. Cover the compact fiber by finitely many coordinate neighborhoods $W_1,\ldots ,W_ k\subset \pi ^{-1}(V_0)$ on which it is integrable. Since $\pi $ is proper, $\pi (Y\setminus (W_1\cup \cdots \cup W_ k))$ is closed and does not contain $x$, so there is a neighborhood $V\subset V_0$ of $x$ with $\pi ^{-1}(V)\subset W_1\cup \cdots \cup W_ k$. The sets $\pi (W_ l\setminus \pi ^{-1}(S))$ cover $V\setminus S$, and $S$ is a null set, so the formula gives $\int _ V|f|^2\mathrm e^{-\varphi }\, \mathrm dV\leq \sum _ l\int _{W_ l}|f\circ \pi |^2|J_\pi |^2\mathrm e^{-\varphi \circ \pi }\, \mathrm dV_ w<\infty $.

Next let $U\subset X$ be open and $h$ a section of $\mathcal O_ Y(K_{Y/X})$ over $\pi ^{-1}(U)$; we may assume that $U$ is connected. Over $U\setminus S$ the map $\pi $ is biholomorphic and $J_\pi $ has no zeros, so $h$ is holomorphic on $\pi ^{-1}(U\setminus S)$, and $h=f\circ \pi $ there for the holomorphic function $f=h\circ (\pi |_{\pi ^{-1}(U\setminus S)})^{-1}$ on $U\setminus S$, which is connected by lemma 9.2.9. If $f\not\equiv 0$, the function $\log |f|^2$ is plurisubharmonic on $U\setminus S$ (proposition 5.10.3), so by corollary 9.3.18 it is locally bounded above near $S$; hence $f$ is locally bounded near $S$, and lemma 9.2.9 extends $f$ holomorphically to $U$. Locally $h=g/J_\pi $ with $g$ holomorphic, and $g=(f\circ \pi )J_\pi $ on the dense open set $\pi ^{-1}(U\setminus S)$, hence everywhere; so $h=f\circ \pi $. The function $f$ is unique, since it is determined on $U\setminus S$, which is dense in $U$.

Finally, let $U\subset X$ be open and $f\in \mathcal O(U)$. Since $\mathcal O_ Y(K_{Y/X})$ is locally generated by $J_\pi ^{-1}$, the function $f\circ \pi $ is a section of $\mathcal O_ Y(K_{Y/X})\mathcal I(\varphi \circ \pi )$ over $\pi ^{-1}(U)$ if and only if $(f\circ \pi )J_\pi $ is a section of $\mathcal I(\varphi \circ \pi )$ there, that is, if and only if (6.12.5) is integrable near every point of $\pi ^{-1}(U)$. By the first part, applied at every point of $U$, this holds if and only if $f$ is a section of $\mathcal I(\varphi )$ over $U$. Together with the second part, which shows that every section of the right side of (6.12.6) over $\pi ^{-1}(U)$ has the form $f\circ \pi $, this proves (6.12.6). $\square$