Lemma 10.5.7C3C3386FA4FC

Proof. Let $A$ be as above. The positive current $\pi ^*\pi _*S$, defined through local potentials as above, agrees with $S$ on $Y\setminus \pi ^{-1}(A)$. Their difference is a closed real current with measure coefficients supported on the proper analytic subset $\pi ^{-1}(A)$ of $Y$. By theorem 9.3.20, it is a finite real linear combination of the currents of integration along the irreducible components of codimension one of the compact analytic set $\pi ^{-1}(A)$, which are exactly the exceptional prime divisors, as shown above; there are finitely many of them by corollary 9.3.16.

It remains to show that the coefficients $c_ E$ are nonnegative. Near a regular point of an exceptional prime divisor $E$ that lies on no other component, the difference is $c_ E[E]=\mathrm{dd}^{\mathrm{c}}(c_ E\log |s_ E|^2)$, where $s_ E$ is a local equation of $E$ (lemma 9.3.19), so local potentials $u'$ of $\pi ^*\pi _*S$ and $u$ of $S$ satisfy $u'=u+c_ E\log |s_ E|^2+p$ with $p$ pluriharmonic. The additivity of Lelong numbers (proposition 6.7.9), applied to this identity if $c_ E\geq 0$ and to $u=u'+|c_ E|\log |s_ E|^2-p$ otherwise, together with $\nu (\log |s_ E|^2,y)=1$, gives $\nu (u',y)=\nu (u,y)+c_ E$ at such points $y$. At points that are moreover generic for both currents, which exist because the points of $E$ excluded by these conditions form a countable union of proper analytic subsets of $E$ (proposition 6.9.7), this reads $\nu (\pi ^*\pi _*S,E)=\nu (S,E)+c_ E$; hence $c_ E=\nu (\pi ^*\pi _*S,E)\geq 0$, since $S$ is non-divisorial.

Since $\pi $ is a biholomorphism over $X\setminus A$, the positive current $\pi _*\pi ^*T$ agrees with $T$ on $X\setminus A$. Hence $\pi _*\pi ^*T-T$ is a closed real current with measure coefficients (theorem 5.3.7) supported on $A$, which has codimension at least two, so it vanishes by theorem 9.3.20. For a prime divisor $E$ on $Y$, the current $\pi _*[E]$ is positive, closed and supported on the analytic set $\pi (E)$. If $E$ is exceptional, $\pi (E)$ has codimension at least two and $\pi _*[E]=0$ by the same theorem. Otherwise $\pi $ maps $E\setminus \pi ^{-1}(A)$ isomorphically onto $\pi (E)\setminus A$, so $\pi _*[E]=[\pi (E)]$ on $X\setminus A$, and the closed current $\pi _*[E]-[\pi (E)]$ of order zero is supported on $\pi (E)\cap A$, of codimension at least two in $X$; it vanishes by the same theorem. $\square$