Theorem 20.4.2A67FA9DFB4BC

Proof. Set $w_ m=(u_ m|_ Y)\circ \pi $. By corollary 5.43.26, these functions are not identically $-\infty $. Moreover, $u_ m|_ Y$ has analytic type $(1/m,\mathfrak a_ m(u;Y))$ by proposition 5.34.1, as in the proof of that corollary, so its pullback $w_ m$ has generic Lelong number $b_ m(E)/m$ along $E$. Thus

Lemma 5.38.1 gives $D_{m,Z}\leq T_ m$. Integration by parts for currents and the Gauduchon condition therefore give

For any fixed finite set of prime divisors, let $m\to \infty $ in the coefficients of these divisors, using theorem 20.3.2, and then take the supremum over all finite sets. This proves (20.4.3) for every finite subsum.

Every prime divisor has strictly positive $\gamma $-area. Hence at most countably many coefficients are positive: for each positive integer $k$, only finitely many terms can have trace mass at least $1/k$, and every positive term is among these for some $k$. Enumerate the corresponding divisors. The finite partial sums are increasing positive closed currents with uniformly bounded trace mass. They therefore converge, in the sense of coefficient measures and weakly as currents, to a positive current of finite mass, which is closed because differentiation is continuous on currents. This constructs $S_ Z(u;Y)$ and proves the integral inequality.

The supremum formula for $\tau _{Y,u}(E)$ gives $b_ m(E)/m\leq \tau _{Y,u}(E)$ for every $m$ and $E$, and the limit along the integers gives coefficientwise convergence. Dominated convergence for the countable series with summable weights $\tau _{Y,u}(E)\int _ E\gamma ^{r-1}$ shows that the trace mass of $S_ Z-D_{m,Z}$ tends to zero. Since a fixed smooth test form is bounded in absolute value, on positive $(1,1)$-currents, by a constant times the trace mass, this gives weak convergence. Finally, $b_{2m}(E)\geq 2b_ m(E)$ gives $D_{2m,Z}\geq D_{m,Z}$. $\square$