Proposition 20.1.19

Proposition 20.1.19.87D0CCEB6DF9 Let $\theta $ be a closed real smooth $(1,1)$-form on $X$, let $(\varphi _ i)_{i\in I}$ be a net in $\mathrm{PSH}(X,\theta )$, and let $\varphi \in \mathrm{PSH}(X,\theta )$. If $\varphi _ i\xrightarrow {d_ S}\varphi $, then

64E2A79320D9 20.1.20

\begin{equation} \mathbb {D}\left(\theta _{\varphi _ i}\right)\to \mathbb {D}\left(\theta _{\varphi }\right). \end{equation}

Proof. Fix a modification $\pi \colon Y\rightarrow X$. Fix a Kähler form $\omega $ on $X$. It suffices to establish the following:

808CB322D92D 20.1.21

\begin{equation} \mathbb {D}\left(\theta _{\varphi _ i}\right)_ Y\to \mathbb {D}\left(\theta _{\varphi }\right)_ Y. \end{equation}

As a map from the pseudometric space $\mathrm{PSH}(X,\theta )$ to the finite-dimensional Euclidean space $\mathrm{H}^{1,1}(Y,\mathbb {R})$, the continuity of $\mathbb {D}(\bullet )_ Y$ can be tested on sequences. Thus, without loss of generality, we may assume that $(\varphi _ i)_ i$ is a sequence and $I=\mathbb {Z}_{>0}$. It is enough to prove that every subsequence has a further subsequence for which (20.1.22) holds.

By corollary 16.9.11, $\pi ^*\varphi _ i\xrightarrow {d_ S}\pi ^*\varphi $. Hence it suffices to prove the claim on $Y$, so we may assume that $\pi $ is the identity. Therefore, we are reduced to the following assertion:

762B9BFFA0ED 20.1.22

\begin{equation} \left\{ \operatorname{Reg}\theta _{\varphi _ i} \right\} \to \left\{ \operatorname{Reg}\theta _{\varphi } \right\} . \end{equation}

After adding a Kähler form to $\theta $, we may also assume that $\theta _{\varphi }$ is a Kähler current. By theorem 16.9.49, replacing all potentials by their $\mathcal{I}$-envelopes does not change the divisorial parts and preserves $d_ S$-convergence. The masses are then bounded from below along a tail, so after passing to a subsequence proposition 16.9.17 supplies increasing and decreasing envelopes $\eta _ i\leq \varphi _ i\leq \psi _ i$ which both converge to $\varphi $ in $d_ S$. Since

\[ \mathbb {D}(\theta _{\eta _ i})_ X\leq \mathbb {D}(\theta _{\varphi _ i})_ X\leq \mathbb {D}(\theta _{\psi _ i})_ X, \]

it is enough to treat the case where $(\varphi _ i)_ i$ is monotone.

Write the divisorial parts of $\theta _{\varphi _ i}$ and $\theta _{\varphi }$ as

\[ \sum _{m=1}^{\infty }a_ m^ i[D_ m],\quad \sum _{m=1}^{\infty }a_ m[D_ m]. \]

By theorem 16.9.55, $a_ m^ i\to a_ m$ for each fixed $m$. If $(\varphi _ i)_ i$ is decreasing, then $a_ m^ i\leq a_ m$ for all $i,m$. The finite-tail argument in the proof of (20.1.18) then gives

\[ \left\{ \operatorname{Reg}\theta _{\varphi }\right\} \leq \left\{ \operatorname{Reg}\theta _{\varphi _ i}\right\} \leq \left\{ \operatorname{Reg}\theta _{\varphi }\right\} +\epsilon \{ \omega \} \]

for all sufficiently large $i$.

If $(\varphi _ i)_ i$ is increasing, then $a_ m\leq a_ m^ i\leq a_ m^1$. We first choose $N$ so that the tails of both $\sum _ m a_ m[D_ m]$ and $\sum _ m a_ m^1[D_ m]$ are at most $(\epsilon /3)\{ \omega \} $ in the pseudo-effective order. The convergence of the finitely many coefficients $a_ m^ i$, $m\leq N$, then gives

\[ \left\{ \operatorname{Reg}\theta _{\varphi _ i}\right\} \leq \left\{ \operatorname{Reg}\theta _{\varphi }\right\} \leq \left\{ \operatorname{Reg}\theta _{\varphi _ i}\right\} +\epsilon \{ \omega \} \]

for all sufficiently large $i$. This proves (20.1.22). $\square$

Proposition 20.1.19`87D0CCEB6DF9`

Comments

Proposition 20.1.1987D0CCEB6DF9

Comments

Proposition 20.1.19`87D0CCEB6DF9`