Proposition 18.26.21.5988333AF401 Let $\omega $ be a Hermitian form on a connected compact complex manifold $X$ of dimension $n\geq 2$. Fix the cutoff $H$, the constant $C$ and the mass $M$ of proposition 18.26.19, and $0<t<1/C$. For every $v\in \mathrm{PSH}(X,\omega )$ with $v\sim tH$, put $v_ k=\max (v,-k)$. The limits
\[ c_ j(v)=\lim _{k\to \infty }\int _ X v_ k(\mathrm{dd}^{\mathrm{c}}v_ k)^{j-1}\wedge \mathrm{dd}^{\mathrm{c}}(\omega ^{n-j}), \qquad 1\leq j<n, \]
exist and are finite, and
\[ \int _ X\mathrm{MA}_\omega (v) =A+\sum _{j=1}^{n-1}\binom nj c_ j(v)-Mt^ n, \qquad A=\int _ X\omega ^ n. \]
In particular, with the lower and upper masses $v_\pm (\omega )$ of definition 13.14.14,
\[ v_-(\omega )-Mt^ n \leq \int _ X\mathrm{MA}_\omega (v) \leq v_+(\omega )-Mt^ n. \]
On a surface this mass is $A+2\int _ Xv\mathrm{dd}^{\mathrm{c}}\omega -Mt^2$.
Proof.
The pole set of $v$ is $S$, and $v$ is locally bounded off $S$. On each relatively compact coordinate neighborhood, choose a fixed smooth local correction $q$ such that $v+q$ and every $v_ k+q$ are plurisubharmonic. By theorem 5.27.18, all closed powers of $\mathrm{dd}^{\mathrm{c}}(v_ k+q)$ converge weakly to finite currents of order zero. Expanding $\mathrm{dd}^{\mathrm{c}}v_ k=\mathrm{dd}^{\mathrm{c}}(v_ k+q)-\mathrm{dd}^{\mathrm{c}}q$ gives weak convergence of $(\mathrm{dd}^{\mathrm{c}}v_ k)^ j$ in every degree $j$. The local limits patch together, because the bounded currents $(\mathrm{dd}^{\mathrm{c}}v_ k)^ j$ are global; write $W_ j$ for the limits.
Integration by parts for bounded potentials gives
\[ \int _ X(\mathrm{dd}^{\mathrm{c}}v_ k)^ j\wedge \omega ^{n-j} =\int _ Xv_ k(\mathrm{dd}^{\mathrm{c}}v_ k)^{j-1} \wedge \mathrm{dd}^{\mathrm{c}}(\omega ^{n-j}); \]
this follows by uniformly bounded decreasing regularization, as in theorem 11.1.5, theorem 5.32.1. Pairing $W_ j$ with the smooth form $\omega ^{n-j}$ shows that $c_ j(v)$ exists and is finite. The top power $(\mathrm{dd}^{\mathrm{c}}v_ k)^ n$ has zero integral for every $k$, by Stokes' theorem for bounded potentials. Expanding the positive bounded measures $(\omega +\mathrm{dd}^{\mathrm{c}}v_ k)^ n$ therefore gives a global positive weak limit $T_ v$ of finite total mass $A+\sum _{j=1}^{n-1}\binom nj c_ j(v)$.
Near $S$, compare $a=v+q$ with $th$; their difference is bounded there. The full top-degree product of $h$ has mass $M$ and is supported on $S$, by theorem 5.27.18, lemma 18.26.15. Hence lemma 5.27.19 gives $(\mathrm{dd}^{\mathrm{c}}a)^ n_{\mathrm{full}}(S)=Mt^ n$. Every closed power of lower degree has zero coefficient mass on $S$, by corollary 5.34.3. In the expansion with the smooth form $\omega -\mathrm{dd}^{\mathrm{c}}q$, only the top power therefore contributes to $T_ v(S)$, and $T_ v(S)=Mt^ n$. Off $S$, the truncations eventually agree with $v$, so $T_ v$ is the non-pluripolar measure of $v$ there. Subtracting the mass on $S$ proves the formula.
Each measure $(\omega +\mathrm{dd}^{\mathrm{c}}v_ k)^ n$ has total mass between $v_-(\omega )$ and $v_+(\omega )$. Passing to the weak limit and subtracting $Mt^ n$ gives the two bounds, the upper one in the extended real sense. In dimension two, $L^1$ convergence of $v_ k$ to $v$ gives $c_1(v)=\int _ Xv\mathrm{dd}^{\mathrm{c}}\omega $, which proves the formula on surfaces.
$\square$
Pome
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