Fix a smooth closed real $(1,1)$-form $\theta $ on $X$.
Definition 16.6.7.E94E1B7F9193 Given $\varphi \in \mathrm{PSH}(X,\theta )$, we define its $\mathcal{I}$-envelope as follows:
FA4D3373FBAE
16.6.8
\begin{equation} P_{\theta }[\varphi ]_{\mathcal{I}}\coloneqq \operatorname *{sup}_{}{}\! {}^{*}\{ \psi \in \mathrm{PSH}(X,\theta ): \psi \leq 0,\psi \sim _{\mathcal{I}} \varphi \} . \end{equation}
If $\varphi =P_{\theta }[\varphi ]_{\mathcal{I}}$, we say $\varphi $ is an $\mathcal{I}$-model potential (in $\mathrm{PSH}(X,\theta )$).
Note that by proposition
5.14.2, $P_{\theta }[\varphi ]_{\mathcal{I}}\in \mathrm{PSH}(X,\theta )$.
Proposition 16.6.9.708B2E9B2EF0 Let $\theta '=\theta +\mathrm{dd}^{\mathrm{c}}g$ for some $g\in C^{\infty }(X)$. Then for any $\varphi \in \mathrm{PSH}(X,\theta )$, we have $\varphi '\coloneqq \varphi -g\in \mathrm{PSH}(X,\theta ')$ and
\[ P_{\theta }[\varphi ]_{\mathcal{I}}\sim P_{\theta '}[\varphi ']_{\mathcal{I}}. \]
Proof.
By symmetry, it suffices to show that
\[ P_{\theta }[\varphi ]_{\mathcal{I}}\preceq P_{\theta '}[\varphi ']_{\mathcal{I}}. \]
After replacing $g$ by $g+C$ for a suitable constant $C$, which only changes $\varphi '$ by an additive constant, we may assume that $g\geq 0$. Let $\psi \in \mathrm{PSH}(X,\theta )$ be such that $\psi \leq 0$ and $\psi \sim _{\mathcal{I}}\varphi $. Then $\psi '\coloneqq \psi -g$ lies in $\mathrm{PSH}(X,\theta ')$, satisfies $\psi '\leq 0$, and is $\mathcal{I}$-equivalent to $\varphi '$ since adding a smooth function does not change multiplier ideals. Hence
\[ \psi -g=\psi '\leq P_{\theta '}[\varphi ']_{\mathcal{I}}. \]
It follows that
\[ \psi -\sup _ X g\leq P_{\theta '}[\varphi ']_{\mathcal{I}}. \]
Taking the upper envelope over all such $\psi $ gives
\[ P_{\theta }[\varphi ]_{\mathcal{I}}-\sup _ X g\leq P_{\theta '}[\varphi ']_{\mathcal{I}}, \]
which proves the desired comparison.
$\square$
Proposition 16.6.10.E535E4DC5384 Let $\pi \colon Y\rightarrow X$ be a proper bimeromorphic morphism from a Kähler manifold $Y$ to $X$. Then for $\varphi \in \mathrm{PSH}(X,\theta )$, we have
\[ P_{\pi ^*\theta }[\pi ^*\varphi ]_{\mathcal{I}}=\pi ^* P_{\theta }[\varphi ]_{\mathcal{I}}. \]
Proof.
Let
\[ \eta =P_{\pi ^*\theta }[\pi ^*\varphi ]_{\mathcal{I}}. \]
First take a candidate $\psi \in \mathrm{PSH}(X,\theta )$ in (16.6.8), so that $\psi \leq 0$ and $\psi \sim _{\mathcal{I}}\varphi $. Then $\pi ^*\psi \leq 0$ and $\pi ^*\psi \sim _{\mathcal{I}}\pi ^*\varphi $ by proposition 16.6.4; hence $\pi ^*\psi \leq \eta $. Taking the upper envelope over all such $\psi $ gives
\[ \pi ^* P_{\theta }[\varphi ]_{\mathcal{I}}\leq \eta . \]
Conversely, let $\chi \in \mathrm{PSH}(Y,\pi ^*\theta )$ be a candidate in the definition of $\eta $, so that $\chi \leq 0$ and $\chi \sim _{\mathcal{I}}\pi ^*\varphi $. By proposition 5.17.8, there is a unique $\psi \in \mathrm{PSH}(X,\theta )$ such that $\chi =\pi ^*\psi $. Since $\pi $ is surjective, $\psi \leq 0$. By proposition 16.6.4, the relation $\chi \sim _{\mathcal{I}}\pi ^*\varphi $ implies $\psi \sim _{\mathcal{I}}\varphi $. Thus $\psi \leq P_{\theta }[\varphi ]_{\mathcal{I}}$, and hence
\[ \chi \leq \pi ^* P_{\theta }[\varphi ]_{\mathcal{I}}. \]
Taking the upper envelope over all such $\chi $ gives the reverse inequality.
$\square$
Proposition 16.6.11.363E28FF4A61 Let $\varphi \in \mathrm{PSH}(X,\theta )$. Then
\[ \varphi \sim _{\mathcal{I}} P_{\theta }[\varphi ]_{\mathcal{I}}. \]
In particular,
\[ P_{\theta }\Bigl [ P_{\theta }[\varphi ]_{\mathcal{I}} \Bigr ]_{\mathcal{I}}=P_{\theta }[\varphi ]_{\mathcal{I}} \]
and the upper semicontinuous regularization in (16.6.8) is not necessary.
Proof.
In view of proposition 16.6.2, it suffices to show that for $k \in \mathbb {Z}_{>0}$, we have
5888DEC3C930
16.6.12
\begin{equation} \mathcal{I}(k\varphi )=\mathcal{I}(kP_{\theta }[\varphi ]_{\mathcal{I}}). \end{equation}
By proposition 5.14.3, we can find $\psi _ i\in \mathrm{PSH}(X,\theta )$ ($i\in \mathbb {Z}_{>0}$) such that $\psi _ i\leq 0$, $\psi _ i\sim _{\mathcal{I}}\varphi $ for all $i\geq 1$ and
\[ \operatorname *{sup}_{i>0}{}\! {}^{*}\psi _ i=P_{\theta }[\varphi ]_{\mathcal{I}}. \]
By proposition 16.6.5, we may replace $\psi _ i$ by $\psi _1\lor \cdots \lor \psi _ i$ and assume that the sequence $\psi _ i$ is increasing. In this case, it follows from the strong openness theorem theorem 5.16.21 that for each $k\in \mathbb {Z}_{>0}$, we have
\[ \mathcal{I}(k\varphi )=\mathcal{I}(k\psi _ j)=\mathcal{I} (kP_{\theta }[\varphi ]_{\mathcal{I}}) \]
for $j$ large enough.
$\square$
Definition 16.6.13.1D9577F4ECC2 Let $\varphi \in \mathrm{PSH}(X,\theta )$, we define the volume $\operatorname{vol}(\theta ,\varphi )$ as
\[ \operatorname{vol}(\theta ,\varphi )=\int _ X \Bigl (\theta +\mathrm{dd}^{\mathrm{c}}P_{\theta }[\varphi ]_{\mathcal{I}} \Bigr )^ n. \]
Proposition 16.6.14.8DF15E9E1928 Let $\theta '=\theta +\mathrm{dd}^{\mathrm{c}}g$ for some $g\in C^{\infty }(X)$. Then for any $\varphi \in \mathrm{PSH}(X,\theta )$, we have $\varphi '=\varphi -g\in \mathrm{PSH}(X,\theta ')$ and
\[ \operatorname{vol}(\theta ,\varphi )= \operatorname{vol}\left(\theta ',\varphi '\right). \]
Proof.
By proposition 16.6.9, the currents $\theta +\mathrm{dd}^{\mathrm{c}}P_\theta [\varphi ]_{\mathcal I}$ and $\theta '+\mathrm{dd}^{\mathrm{c}}P_{\theta '}[\varphi ']_{\mathcal I}$ have the same singularity type in the same cohomology class. Applying the monotonicity theorem theorem 13.13.11 gives equality of their non-pluripolar masses.
$\square$
In view of proposition 16.6.14, the volume $\operatorname{vol}(\theta ,\varphi )$ depends only on the current $\theta _{\varphi }$, and we may write
3535B0853CBC
16.6.15
\begin{equation} \operatorname{vol}\theta _{\varphi }=\operatorname{vol}(\theta ,\varphi ). \end{equation}
Definition 16.6.16.202451F22A2A Let $\alpha \in \mathrm{H}^{1,1}(X,\mathbb {R})$ be a pseudo-effective class. The volume $\operatorname{vol}\alpha $ of $\alpha $ is defined as
\[ \operatorname{vol}\alpha =\operatorname{vol}T_{\min }, \]
where $T_{\min }$ is a current with minimal singularities in $\alpha $. Note that $\operatorname{vol}\alpha $ is independent of the choice of $T_{\min }$, thanks to theorem 13.13.11.
Let us recall the following elementary result for later use.
Proposition 16.6.17.12DD9B82F232 Let $\pi \colon Y\rightarrow X$ be a proper bimeromorphic morphism from a Kähler manifold $Y$.
7C6266FB792D
For any big class $\alpha \in \mathrm{H}^{1,1}(X,\mathbb {R})$, $\pi ^*\alpha $ is big. Moreover, $\operatorname{vol}\pi ^*\alpha =\operatorname{vol}\alpha $.
FDD4AF241C61
For any big class $\beta \in \mathrm{H}^{1,1}(Y,\mathbb {R})$, $\pi _*\beta $ is big. Moreover, $\operatorname{vol}\pi _*\beta \geq \operatorname{vol}\beta $.
Proof.
1 Take a current $T_{\min }$ with minimal singularities in $\alpha $. Then $\pi ^*T_{\min }$ is a current with minimal singularities in $\pi ^*\alpha $. Our assertion follows from proposition 13.13.4.
2 Let $S_{\min }$ be a current with minimal singularities in $\beta $. Then $\pi _*S_{\min }$ is a positive current in the cohomology class $\pi _*\beta $. A current with minimal singularities in $\pi _*\beta $ is less singular than $\pi _*S_{\min }$, hence by the monotonicity theorem theorem 13.13.11
\[ \operatorname{vol}(\pi _*\beta )\geq \int _ X\left(\pi _*S_{\min }\right)^ n. \]
By proposition 13.13.4,
\[ \int _ X \left(\pi _*S_{\min }\right)^ n\geq \int _ Y S_{\min }^ n=\operatorname{vol}\beta . \]
We conclude $\operatorname{vol}\pi _*\beta \geq \operatorname{vol}\beta $.
$\square$
The $\mathcal{I}$-envelope and the $P$-envelope are related in a simple manner.
Proposition 16.6.18.8AE79B76D6D2 Let $\varphi \in \mathrm{PSH}(X,\theta )$. Then
\[ P_{\theta }[\varphi ]\leq P_{\theta }[\varphi ]_{\mathcal{I}},\quad \varphi \sim _{\mathcal{I}}P_{\theta }[\varphi ]. \]
As a consequence,
0BC2A1135095
16.6.19
\begin{equation} P_{\theta }\Bigl [P_{\theta }[\varphi ]\Bigr ]_{\mathcal{I}}=P_{\theta }[\varphi ]_{\mathcal{I}}. \end{equation}
Proof.
It suffices to show that $\varphi \sim _{\mathcal{I}} P_{\theta }[\varphi ]$. Namely, for each $k\in \mathbb {Z}_{>0}$, we have
EA49AC910A56
16.6.20
\begin{equation} \mathcal{I}(k\varphi )=\mathcal{I}\Bigl (kP_{\theta }[\varphi ]\Bigr ). \end{equation}
Fix $k$ for now. It follows from (16.3.7) and the strong openness theorem theorem 5.16.21 that
\[ \mathcal{I}\Bigl (kP_{\theta }[\varphi ]\Bigr )=\mathcal{I}\Bigl ((k\varphi +C)\land kV_{\theta }\Bigr ), \]
when $C$ is large enough. Since $(k\varphi +C)\land kV_{\theta }\sim k\varphi $, we have
\[ \mathcal{I}\Bigl ((k\varphi +C)\land kV_{\theta }\Bigr )=\mathcal{I}(k\varphi ) \]
and (16.6.20) follows.
$\square$
In particular, we obtain an interesting relation between the non-pluripolar mass and the volume.
Corollary 16.6.21.FB21F0385323 Let $\varphi \in \mathrm{PSH}(X,\theta )$. Then
\[ \int _ X \theta _{\varphi }^ n\leq \operatorname{vol}\theta _{\varphi }. \]
The reverse inequality fails in general, see example
16.8.19.
Proof.
This follows from proposition 16.6.18, theorem 13.13.11 and proposition 16.3.10.
$\square$
We note the following special case:
Proposition 16.6.22.10F9DDE25703 Let $\varphi \in \mathrm{PSH}(X,\theta )$. Assume that $\varphi $ has analytic singularities. Then
B621808C9E65
16.6.23
\begin{equation} \varphi \sim P_{\theta }[\varphi ] \sim P_{\theta }[\varphi ]_{\mathcal{I}}. \end{equation}
Hence, $P_{\theta }[\varphi ]$ and $P_{\theta }[\varphi ]_{\mathcal{I}}$ both have analytic singularities.
In particular,
D91D171A0296
16.6.24
\begin{equation} \int _ X \theta _{\varphi }^ n= \operatorname{vol}\theta _{\varphi }. \end{equation}
Proof.
First observe that (16.6.24) follows from (16.6.23) and theorem 13.13.11. It remains to establish (16.6.23).
In view of proposition 16.6.18, it suffices to show that
9169FBAAFD41
16.6.25
\begin{equation} P_{\theta }[\varphi ]_{\mathcal{I}}\preceq \varphi . \end{equation}
By proposition 16.6.10, proposition 16.4.1 and theorem 5.18.8, after passing to a log resolution we may assume that $\varphi $ has log singularities along an effective $\mathbb Q$-divisor $D$. This reduction does not change the assertion, since both the $P$- and the $\mathcal I$-envelope are compatible with bimeromorphic pull-back.
After multiplying $\theta $ and $\varphi $ by a sufficiently divisible positive integer, and then dividing the final comparison by the same integer, we may assume that $D$ is integral. In this case the Bergman-type quasi-equisingular approximants of $\varphi $ associated to the ideals $\mathcal I(2^ j\varphi )$ have the same singularity type as $\varphi $.
Since
\[ P_\theta [\varphi ]_{\mathcal I}\sim _{\mathcal I}\varphi \]
by proposition 16.6.11, we have
\[ \mathcal I(2^ jP_\theta [\varphi ]_{\mathcal I})=\mathcal I(2^ j\varphi ). \]
Thus the quasi-equisingular approximants $\eta _ j$ of $P_\theta [\varphi ]_{\mathcal I}$ and $\varphi _ j$ of $\varphi $ may be chosen with the same singularity type $(2^{-j},\mathcal I(2^ j\varphi ))$. Consequently $\eta _ j\sim \varphi _ j\sim \varphi $ for $j\gg 1$. Since $\eta _ j\searrow P_\theta [\varphi ]_{\mathcal I}$, choosing one such large $j$ gives
\[ P_\theta [\varphi ]_{\mathcal I}\leq \eta _ j\leq \varphi +C_ j, \]
and hence (16.6.25) holds.
$\square$
Pome
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