Pome

Proposition 16.21.2037CA226D141

Proposition 16.21.2.037CA226D141 Let $u\in \mathrm{PSH}(X,\theta )$.

  1. $P_\theta [u]$ depends only on the singularity type of $u$, and $u\preceq u'$ implies $P_\theta [u]\leq P_\theta [u']$.

  2. The rooftop envelopes $P_\theta (u+k,V_\theta )$ increase with $k$, and

    \[ P_\theta [u]=\left(\lim _{k\to \infty } P_\theta (u+k,V_\theta )\right)^*. \]

  3. The singularity envelope preserves all mixed non-pluripolar masses: if $\theta _1,\ldots ,\theta _ p$, $1\leq p\leq n$, are smooth closed real $(1,1)$-forms and $u_ j\in \mathrm{PSH}(X,\theta _ j)$, then

    \[ \int _ X\left\langle \bigwedge _ j(\theta _ j+\mathrm{dd}^{\mathrm{c}}P_{\theta _ j}[u_ j])\right\rangle \wedge \omega ^{n-p} =\int _ X\left\langle \bigwedge _ j(\theta _ j+\mathrm{dd}^{\mathrm{c}}u_ j)\right\rangle \wedge \omega ^{n-p}. \]

    In particular, $P_\theta [u]$ has the same mass as $u$.

  4. If $u$ has minimal singularities, then $P_\theta [u]=V_\theta $.

Proof. (1) The family of competitors depends only on the singularity type of $u$, and it increases when $u$ is replaced by a less singular potential.

(2) The rooftop envelopes increase with $k$ because the rooftop envelope is monotone. Every competitor $v$ for $P_\theta [u]$ satisfies $v\leq u+k$ for some integer $k$ and $v\leq V_\theta $, hence $v\leq P_\theta (u+k,V_\theta )$. Conversely, each $P_\theta (u+k,V_\theta )$ is nonpositive and, by proposition 16.11.6, has the singularity type of $u$; so it is a competitor. Hence the two families have the same supremum, which proves the formula.

(3) By proposition 16.11.6, the tuple of rooftop envelopes $P_{\theta _ j}(u_ j+k,V_{\theta _ j})$ has the singularity types of the $u_ j$, so it has the same mixed mass as the tuple $(u_ j)$ by corollary 13.8.8. By (2) and theorem 5.10.2, which shows that an increasing sequence differs from its regularized limit only on a Lebesgue null set, these rooftop envelopes increase almost everywhere to $P_{\theta _ j}[u_ j]$ as $k\to \infty $, and corollary 13.8.9 shows that their mixed masses converge to the mixed mass of the envelopes.

(4) If $u$ has minimal singularities, then $V_\theta \preceq u$, so $V_\theta $ is itself a competitor; it dominates all other competitors, since they are nonpositive. $\square$

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