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20.3.32 Exact realization on a Kähler submanifold4C8D52A64B1A

By theorem 20.3.27, semicontinuity of Lelong numbers gives $\nu (\mathcal T_\Omega (u;Y)\circ \pi ,E)\geq \tau _{Y,u}(E)$. When $Y$ is Kähler, positive mass in a sufficiently large closed background strengthens this to equality: for decreasing sequences of model potentials whose limit has positive mass, the divisorial Lelong numbers converge. The envelope obtained in this way then gives the reverse inequality in every larger background.

Theorem 20.3.33.D1551EE7322B Suppose $Y$ is Kähler. Choose a Hermitian form $\Omega _0\geq \theta |_ Y$ and a Kähler form $\kappa \geq 2\Omega _0$ on $Y$. For every Hermitian form $\Omega \geq \kappa $,

\[ \nu \bigl (\mathcal T_\Omega (u;Y)\circ \pi ,E\bigr ) =\tau _{Y,u}(E) \]

for every prime divisor $E$ on every smooth compact modification $\pi :Z\to Y$.

Proof. The curvature errors $\epsilon _ m$ tend to zero, so $u_ m|_ Y\in \mathrm{PSH}(Y,\kappa )$ for all sufficiently large $m$. By theorem 20.3.27, applied with $\Omega =\kappa $ and $\delta _ m=0$, the envelope fixed points

\[ q_ j=P_\kappa [u_{2^ j}|_ Y] \]

decrease to $T=\mathcal T_\kappa (u;Y)$, after discarding finitely many indices. The normalized potential $v$ of proposition 20.3.24 belongs to $\mathrm{PSH}(Y,\Omega _0)\subset \mathrm{PSH}(Y,\kappa /2)$ and satisfies $v\leq T$, so $P_{\kappa /2}(T)\not\equiv -\infty $. Thus lemma 16.28.69 and Stokes' theorem give, with $r=\dim Y$,

\[ \int _ Y\mathrm{MA}_\kappa (T) \geq 2^{-r}\int _ Y\kappa ^ r>0. \]

For a decreasing sequence of envelope fixed points in a closed background, positive mass of the limit implies convergence of every divisorial Lelong number [Proposition 3.1.12, SGPT]. Applied on $Y$, this yields

\[ \nu (T\circ \pi ,E) =\lim _ j\nu (q_ j\circ \pi ,E) =\lim _ j2^{-j}\operatorname {ord}_ E\mathfrak a_{2^ j}(u;Y) =\tau _{Y,u}(E). \]

Here the middle equality uses the ordinary equivalence of each $q_ j$ with $u_{2^ j}|_ Y$.

If $\Omega \geq \kappa $, then $T$ is a trace competitor for $\Omega $, so $T\leq T_\Omega =\mathcal T_\Omega (u;Y)$. Together with the defining inequalities of $T_\Omega $, this gives

\[ \tau _{Y,u}(E)\leq \nu (T_\Omega \circ \pi ,E) \leq \nu (T\circ \pi ,E)=\tau _{Y,u}(E). \]

This proves the assertion for every such $\Omega $. $\square$

Only $Y$ is required to be Kähler here; the closed theory is applied on $Y$. If $Y$ has the bounded mass property, corollary 13.21.5 gives the same exactness in a sufficiently large Hermitian background with positive lower mass. For a general non-Kähler $Y$, the envelope limit gives only the lower bound on its divisorial Lelong numbers.

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