13.6 Fiber integration and modificationsFBE95DA26374

Proof. In product charts choose smooth local potentials $a(x)$ of $\theta $ and $b(z)$ of $\gamma $. Then $W+a+b$ is plurisubharmonic, so by proposition 5.9.3 its restriction $W_ z+a+b(z)$ to a fiber is plurisubharmonic in $x$ or identically $-\infty $; the second alternative is excluded, because $W_ z\geq \varphi -C$ and $\varphi $ is not identically $-\infty $ on any open set. Hence $W_ z$ is quasi-plurisubharmonic with $\theta +\mathrm{dd}^{\mathrm{c}}W_ z\geq 0$, that is, $W_ z\in \mathrm{PSH}(X,\theta )$. All the products in (13.6.2) are finite by theorem 13.5.4, and (13.5.5) bounds the fiber masses uniformly in $z$, because the shifted background can be chosen from the fixed form $\theta $, independently of $W_ z$.

Put $w=W+a+b$, $v=\Phi +a+b$ and $O_ k=\{ w>-k\} \cap \{ v>-k\} $ in a product chart. Let $\rho \geq 0$ be a smooth function with compact support in the chart, let $G\geq 0$ be bounded and Borel, and let $\eta \geq 0$. Apply (13.4.6) to the bounded functions $\max \{ w,-k\} $ and $\max \{ v,-k\} $, with multiplicities $j$ and $r-j$, and to the weight $\rho G\mathbf1_{O_ k}$. The set $O_ k$ is contained in the level-$k$ set of the tuple formed by $j$ copies of $w$ and $r-j$ copies of $v$ (if $j=0$ or $j=r$, one of the two conditions defining $O_ k$ concerns a potential that does not occur in the tuple). Hence, by lemma 13.2.2, the restriction to $O_ k$ of the bounded mixed product equals the restriction to $O_ k$ of the non-pluripolar product, on the total space and, with $(O_ k)_ z$ in place of $O_ k$, on every fiber. These restrictions increase with $k$, and they increase to the non-pluripolar products, because the complements of $\bigcup _ kO_ k$ and of $\bigcup _ k(O_ k)_ z$ are contained in the pole sets of $w,v$ and of $w_ z,v_ z$, which are pluripolar and carry no mass. Monotone convergence, on the total space and in $z$, therefore gives the fiber formula for the non-pluripolar products with weight $\rho G$.

Taking $G=1$ gives the unweighted identity. Since $\Phi -C\leq W\leq \Phi +C$, the potentials $W$ and $\Phi $ have the same pole set, and so do $W_ z$ and $\varphi $; take $G=W-\Phi +C\in [0,2C]$ off this set and $G=C$ on it, and subtract $C$ times the unweighted identity. Summing over a finite partition of unity by such functions $\rho $ (proposition 5.2.6), and writing a signed $\eta $ as a difference of nonnegative forms, gives (13.6.2). All integrals are finite by the uniform mass bound and $|W-\Phi |\leq C$, and the values assigned on the common pole set do not matter, because this set is pluripolar and carries no mass for any of the products. $\square$

Proof. The pullback $\pi ^*T_ j$ is the closed positive current of proposition 5.6.5, defined by pulling back local plurisubharmonic potentials; a pulled-back potential is not identically $-\infty $ on any open set, because $\pi $ is biholomorphic over a dense open set whose preimage is dense (proposition 9.8.3 and the definition of bimeromorphic maps in section 9.4). The pushforward $\pi _*R$ of a closed positive current is closed and positive, since pullback of smooth forms commutes with $\mathrm d$ and preserves strong positivity (proposition 5.3.2). All products in the statement are therefore defined, and they are finite by theorem 13.5.4.

Choose a nowhere dense analytic subset $A\subset X$ such that $\pi ^{-1}(A)$ is nowhere dense and $\pi $ restricts to a biholomorphism $\pi ^{-1}(X\setminus A)\to X\setminus A$. Both $A$ and $\pi ^{-1}(A)$ are pluripolar: near each point, a nowhere dense analytic set is contained in the zero set of a holomorphic function $f$ that does not vanish identically on a connected neighborhood, hence in the pole set of $\log |f|^2$. For a point $\widetilde x\in \pi ^{-1}(X\setminus A)$, use on a neighborhood of $\widetilde x$ the coordinates $z\circ \pi $, where $z$ are coordinates near $\pi (\widetilde x)$. In these coordinates $\pi $ is the identity, and the local potentials $u_ j\circ \pi $ of $\pi ^*T_ j$ are the local potentials $u_ j$ of $T_ j$. Since non-pluripolar products do not depend on the choice of coordinates or of local potentials, the two sides of the first identity agree on $X\setminus A$.

It remains to show that neither side charges $A$. The right side gives the pluripolar set $A$ zero mass. The product $S=\langle \pi ^*T_1\wedge \cdots \wedge \pi ^*T_ p\rangle $ gives $\pi ^{-1}(A)$ zero mass, and hence $\pi _*S$ gives $A$ zero mass. Indeed, in a chart of $X$, the trace measure of $\pi _*S$ is the image under $\pi $ of the positive measure $S\wedge \pi ^*\beta ^{n-p}$, which does not charge $\pi ^{-1}(A)$, and the coefficient measures of $\pi _*S$ are bounded by a multiple of this trace measure (theorem 5.3.7). Hence the two sides of the first identity have the same coefficient measures on $X\setminus A$ and on $A$, and they are equal. The same argument on $X\setminus A$, where now the local potentials of $\pi _*R_ j$ are those of $R_ j$ in the coordinates $z\circ \pi $, proves the second identity. In top degree, the total mass of $\pi _*S$ is $(\pi _*S)(1)=S(\pi ^*1)=S(1)$, which proves the assertion on masses. $\square$