Proposition 18.26.196396018F7822

Proof. Outside a fixed neighborhood of $S$, the truncations $h_ L$ eventually agree with $h$ on every compact set, in particular on the supports of the derivatives of $\chi $. The cutoff calculation in proposition 18.26.17 therefore gives a lower bound $-C\omega $ for $\mathrm{dd}^{\mathrm{c}}H_ L$ and $\mathrm{dd}^{\mathrm{c}}H$, uniform in $L$.

Let $\kappa =\mathrm{dd}^{\mathrm{c}}|z|^2$ on the coordinate polydisk $D$, and choose $B>0$ so that $T_ L=B\kappa +\mathrm{dd}^{\mathrm{c}}H_ L$ is positive and closed on $D$. The potentials of $T_ L$ are locally bounded. Since $H_ L$ has compact support, integration by parts for bounded potentials gives, for $1\leq j\leq n$,

Indeed, every nonconstant term of the binomial expansion is the complex Hessian of a compactly supported current built from bounded products, wedged with a closed power of $\kappa $. Thus the traces in all degrees are bounded independently of $L$.

For $j<n$, every subsequential weak limit of $T_ L^ j$ has coefficient measures vanishing on $S$, by corollary 5.34.3. Off $S$, local boundedness and eventual equality identify it with $(B\kappa +\mathrm{dd}^{\mathrm{c}}H)^ j$. These two properties determine the limit uniquely. Expanding $(T_ L-B\kappa )^ j$ therefore shows that $(\mathrm{dd}^{\mathrm{c}}H_ L)^ j$ converges weakly to a finite signed current $W_ j$ with no coefficient mass on $S$. It vanishes near $\partial D$ and extends by zero to $X$. Integration by parts for bounded potentials gives

The left side converges to $\int _ XW_ j\wedge \omega ^{n-j}$, which proves that $c_ j$ exists and is finite.

Fix $0<t<1/C$. The positive measures $(\omega +t\mathrm{dd}^{\mathrm{c}}H_ L)^ n$ have uniformly bounded mass, by their local expansion in powers of $T_ L$ and smooth forms. Their total masses satisfy

since the top power $(\mathrm{dd}^{\mathrm{c}}H_ L)^ n$ integrates to zero by Stokes' theorem for bounded potentials. Let $\eta $ be a weak subsequential limit. Its total mass is $A+\sum _{j=1}^{n-1}\binom nj c_ jt^ j$.

Near $S$, where $\chi =1$, expand $(\omega +t\mathrm{dd}^{\mathrm{c}}h_ L)^ n$. All the limits of lower degree give $S$ zero mass, while the top-degree measures have mass $M$ and supports converging to $S$, by lemma 18.26.15. Hence $\eta (S)=Mt^ n$. Off $S$, eventual equality with the locally bounded function $tH$ identifies $\eta $ with the non-pluripolar measure of $tH$. Subtracting the mass on $S$ proves the formula.

If $\mathrm{dd}^{\mathrm{c}}(\omega ^ q)=0$ for $1\leq q<n$, every $c_ j$ is zero. For $n=2$, dominated $L^1$ convergence $H_ L\to H$ gives $c_1=\int _ XH\mathrm{dd}^{\mathrm{c}}\omega $, which proves the formula on surfaces. These formulas give the masses of the potentials $tH$ themselves. Passing to singularity volumes in a nonclosed background requires control of the masses of all potentials ordinarily equivalent to $tH$. $\square$