Proposition 13.6.30811FFDF15B7

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. $\square$