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20.2.38 Restricted volumes of line bundlesEDE927CBCEA7

Let $X$ be a connected projective manifold of dimension $n$ and $Y\subseteq X$ be a connected submanifold of dimension $m$. Consider a big line bundle $L$ on $X$, a Hermitian metric $h_0$ on $L$ with $\theta =c_1(L,h_0)$. Let $A$ be a very ample line bundle on $X$. Take a Hermitian metric $h_ A$ on $A$ such that $\omega =\mathrm{dd}^{\mathrm{c}}h_ A$ is a Kähler form.

Using the trace operator, one could prove the following generalization of theorem 18.1.46:

Theorem 20.2.39.230AF20F8E84 Let $h$ be a (singular) plurisubharmonic metric on $L$ with $\nu (\mathrm{dd}^{\mathrm{c}}h,Y)=0$. Assume that

92546FA16294 20.2.40
\begin{equation} \lim _{\epsilon \to 0+} \int _ Y\left(\operatorname{Tr}_ Y^{c_1(L|_ Y)+\epsilon \omega |_ Y}\bigl (c_1(L,h)\bigr ) \right)^ m>0. \end{equation}

Then for any holomorphic line bundle $T$ on $X$ we have

97B6F891A18A 20.2.41
\begin{equation} \int _ Y \left(\operatorname{Tr}_ Y^{c_1(L|_ Y)}\bigl (c_1(L,h)\bigr )\right)^ m = \lim _{k \to \infty } \frac{m!}{k^ m} h^0\left(Y,T|_ Y\otimes L|_ Y^ k \otimes \operatorname{Res}_ Y\left(\mathcal{I}(h^{k})\right) \right). \end{equation}
Recall that $\operatorname{Res}_ Y$ is defined in definition 5.16.23. Observe that by example 20.2.17, (20.2.40) implies that $\operatorname{Tr}_ Y^{c_1(L|_ Y)}(c_1(L,h))$ is defined. So (20.2.41) is defined.

We will identify $h$ with $\varphi \in \mathrm{PSH}(X,\theta )$ as in (5.20.9).

For the proof, recall that the ideal sheaf $\mathcal{I}_{\infty }$ is introduced in definition 5.18.13.

Proof. Step 1. We first assume that $\mathrm{dd}^{\mathrm{c}}h$ is a Kähler current. Let $(\varphi _ j)_ j$ be a quasi-equisingular approximation of $\varphi $ in $\mathrm{PSH}(X,\theta )$. After possibly replacing $(\varphi _ j)_ j$ by a subsequence, there exists $\epsilon _0 \in (0,1)\cap \mathbb {Q}$ such that the $\theta _{(1-\epsilon ) \varphi _ j}$'s are also Kähler currents for any $\epsilon \in (0,\epsilon _0)$.

For any rational $\epsilon \in (0,\epsilon _0)$, and sufficiently large $k$, we have

\[ \begin{aligned} & \frac{1}{m!}\int _ Y \left(\theta |_ Y+\mathrm{dd}^{\mathrm{c}}\operatorname{Tr}_ Y^{\theta }(\varphi )\right)^ m \\ =& \lim _{k \to \infty } \frac{1}{k^ m} h^0\left(Y, T|_ Y\otimes L|_ Y^ k\otimes \mathcal{I}\left(k\operatorname{Tr}_ Y^{\theta }(\varphi )\right)\right) \quad \textup{by theorem \href{../../tag/FEA63C5AE11F/}{18.1.46}}\\ \leq & \varliminf _{k \to \infty } \frac{1}{k^ m} h^0\left(Y, T|_ Y\otimes L|_ Y^ k\otimes \operatorname{Res}_ Y\left(\mathcal{I}(k\varphi )\right)\right) \quad \textup{ by proposition \href{../../tag/DB90B2D3ADEB/}{20.2.26}}\\ \leq & \varlimsup _{k \to \infty } \frac{1}{k^ m} h^0\left(Y, T|_ Y\otimes L|_ Y^ k\otimes \operatorname{Res}_ Y\left(\mathcal{I}(k\varphi )\right)\right)\\ \leq & \varlimsup _{k \to \infty } \frac{1}{k^ m} h^0\left(Y, T|_ Y\otimes L|_ Y^ k\otimes \operatorname{Res}_ Y\mathcal{I}\left(k\varphi _ j\right)\right) \\ \leq & \varlimsup _{k \to \infty } \frac{1}{k^ m} h^0\left(Y, T|_ Y\otimes L|_ Y^ k\otimes \operatorname{Res}_ Y\mathcal{I}_{\infty }\left((1-\epsilon )k\varphi _ j\right)\right) \quad \textup{by \href{../../tag/2E0432C779A0/}{5.18.15}}\\ \leq & \varlimsup _{k \to \infty } \frac{1}{k^ m} h^0\left(Y, T|_ Y\otimes L|_ Y^ k\otimes \mathcal{I}_{\infty }\left((1-\epsilon )k\varphi _ j|_ Y\right)\right)\\ \leq & \varlimsup _{k \to \infty } \frac{1}{k^ m} h^0\left(Y, T|_ Y\otimes L|_ Y^ k\otimes \mathcal{I}\left((1-\epsilon ) k\varphi _ j|_ Y\right) \right)\\ = & \frac{1}{m!} \int _ Y \left(\theta |_ Y+(1-\epsilon )\mathrm{dd}^{\mathrm{c}}\varphi _ j|_ Y\right)^ m\quad \textup{by theorem \href{../../tag/FEA63C5AE11F/}{18.1.46}}. \end{aligned} \]

Here we used the observation that for any $\psi \in \mathrm{QPSH}(X)$ with analytic singularities with $\psi |_ Y\not\equiv -\infty $, we have

\[ \operatorname{Res}_ Y \mathcal{I}_{\infty }(\psi )\subseteq \mathcal{I}_{\infty }\left(\psi |_ Y\right). \]

Letting $\epsilon \to 0+$, 16.9.25 gives

\[ (1-\epsilon )\varphi _ j|_ Y\xrightarrow {d_ S}\varphi _ j|_ Y. \]

Hence, by theorem 16.9.30, the last term converges to

\[ \frac{1}{m!}\int _ Y \left(\theta |_ Y+\mathrm{dd}^{\mathrm{c}}\varphi _ j|_ Y\right)^ m. \]

We conclude that

\[ \begin{aligned} \frac{1}{m!}\int _ Y \left(\theta |_ Y+\mathrm{dd}^{\mathrm{c}}\operatorname{Tr}_ Y^{\theta }(\varphi )\right)^ m\leq & \varliminf _{k \to \infty } \frac{1}{k^ m} h^0\left(Y, T|_ Y\otimes L|_ Y^ k\otimes \operatorname{Res}_ Y\left(\mathcal{I}(k\varphi )\right)\right) \\ \leq & \varlimsup _{k \to \infty } \frac{1}{k^ m} h^0\left(Y, T|_ Y\otimes L|_ Y^ k\otimes \operatorname{Res}_ Y\left(\mathcal{I}(k\varphi )\right)\right)\\ \leq & \frac{1}{m!} \int _ Y \left(\theta |_ Y+\mathrm{dd}^{\mathrm{c}}\varphi _ j|_ Y\right)^ m. \end{aligned} \]

Letting $j\to \infty $, we conclude (20.2.41).

Step 2. We handle the general case. After adding a constant to $\varphi $, we may assume that $\varphi \leq 0$.

Using proposition 20.2.26 and theorem 18.1.46 we obtain that

\[ \begin{aligned} \int _ Y \left(\theta |_ Y+\mathrm{dd}^{\mathrm{c}}\operatorname{Tr}_ Y^{\theta }(\varphi )\right)^ m & = \lim _{k \to \infty } \frac{m!}{k^ m} h^0\left(Y,T|_ Y \otimes L|_ Y^ k \otimes \mathcal{I}\left(k\operatorname{Tr}_ Y^{\theta }(\varphi )\right)\right)\\ & \leq \varliminf _{k \to \infty } \frac{m!}{k^ m} h^0\left(Y,T|_ Y \otimes L|_ Y^ k \otimes \operatorname{Res}_ Y\left(\mathcal{I}(k\varphi )\right)\right). \end{aligned} \]

Now we address the other direction in (20.2.41). Let $\phi \in \mathrm{H}^0(X,A)$ be a section that does not vanish identically on $Y$. Such $\phi $ exists since $A$ is very ample.

We fix $k_0 \in \mathbb {N}$. For any $k \geq 0$, we have $k = q k_0 + r$ with $q,r \in \mathbb {N}$ and $r \in \{ 0,\ldots , k_0-1\} $. Also, we have an injective linear map

\[ \mathrm{H}^0\left(Y,T|_ Y \otimes L|_ Y^ k \otimes \operatorname{Res}_ Y\mathcal{I}\left(k \varphi \right)\right)\xrightarrow {\cdot \phi ^{\otimes q}} \mathrm{H}^0\left(Y,T|_ Y\otimes L|_ Y^ k \otimes A|_ Y^ q \otimes \operatorname{Res}_ Y\mathcal{I}\left(k \varphi \right)\right). \]

Therefore,

\[ \begin{aligned} & \varlimsup _{k\to \infty } \frac{m!}{k^ m} h^0\left(Y,T|_ Y \otimes L|_ Y^ k \otimes \operatorname{Res}_ Y\mathcal{I}\left(k\varphi \right)\right) \\ \leq & \varlimsup _{k\to \infty } \frac{m!}{k^ m} h^0\left(Y,T|_ Y \otimes L|_ Y^ k \otimes A|_ Y^ q\otimes \operatorname{Res}_ Y\mathcal{I}\left(k\varphi \right)\right) \\ \leq & \max _{r=0,\ldots ,k_0-1}\frac{1}{k_0^ m} \varlimsup _{q\to \infty } \frac{m!}{q^ m} h^0\left(Y,T|_ Y\otimes L|_ Y^{q k_0} \otimes A|_ Y^ q\otimes L|_ Y^{r}\otimes \operatorname{Res}_ Y\mathcal{I}\left((qk_0+r)\varphi \right)\right)\\ \leq & \max _{r=0,\ldots ,k_0-1}\frac{1}{k_0^ m} \varlimsup _{q\to \infty } \frac{m!}{q^ m} h^0\left(Y,T|_ Y\otimes L|_ Y^{q k_0} \otimes A|_ Y^ q\otimes L|_ Y^{r}\otimes \operatorname{Res}_ Y\mathcal{I}\left(k_0q \varphi \right)\right)\\ = & \int _ Y \left(\theta |_ Y+k_0^{-1}\omega |_ Y+\mathrm{dd}^{\mathrm{c}}\operatorname{Tr}_ Y^{\theta +k_0^{-1}\omega }(\varphi )\right)^ m\\ =& \int _ Y \left(\theta |_ Y+k_0^{-1}\omega |_ Y+\mathrm{dd}^{\mathrm{c}}\operatorname{Tr}_ Y^{\theta }(\varphi )\right)^ m, \end{aligned} \]

where the maximum is taken over the finitely many residue classes $k=qk_0+r$, the following inequality uses $k_0q\leq k$, the first equality follows by applying Step 1 for each $r$ to the big line bundle $L^{k_0} \otimes A$, the Kähler current $k_0 \theta _{\varphi } + \omega $, and twisting bundle $T \otimes L^ r$, and the second equality follows from proposition 16.3.10. Letting $k_0\to \infty $ and using the multi-linearity of the non-pluripolar product proposition 13.11.4 6, we conclude that

\[ \varlimsup _{k\to \infty } \frac{m!}{k^ m} h^0\left(Y,T|_ Y \otimes L|_ Y^ k \otimes \operatorname{Res}_ Y\mathcal{I}\left(k \varphi \right)\right)\leq \int _ Y \left(\theta |_ Y+\mathrm{dd}^{\mathrm{c}}\operatorname{Tr}_ Y^{\theta }(\varphi )\right)^ m. \]

This completes the proof. $\square$

Theorem 20.2.42.4F486DDB8BD0 Let $\varphi \in \mathrm{PSH}(X,\theta )$ such that $\nu (\varphi ,Y) =0$. Assume that $\theta _{\varphi }$ is a Kähler current. Then for any holomorphic line bundle $T$ on $X$ we have

\[ \int _ Y \left(\theta |_ Y+\mathrm{dd}^{\mathrm{c}}\operatorname{Tr}_ Y^{\theta }(\varphi )\right)^ m = \lim _{k \to \infty } \frac{m!}{k^ m} \dim _{\mathbb C}\left\{ s|_ Y: s\in \mathrm{H}^0\left(X,T\otimes L^ k \otimes \mathcal{I}(k\varphi )\right)\right\} . \]

Proof. This is a consequence of theorem 18.1.46, theorem 20.2.28 and theorem 20.2.39:

\[ \begin{aligned} \int _ Y \left(\theta |_ Y+\mathrm{dd}^{\mathrm{c}}\operatorname{Tr}_ Y^{\theta }(\varphi )\right)^ m & = \lim _{k \to \infty } \frac{m!}{k^ m} h^0\left(Y,T|_ Y\otimes L|_ Y^ k\otimes \mathcal{I}\left(k \operatorname{Tr}_ Y^{\theta }(\varphi )\right)\right)\\ & \leq \varliminf _{k \to \infty } \frac{m!}{k^ m} \dim _{\mathbb C}\left\{ s|_ Y: s\in \mathrm{H}^0\left(X,T\otimes L^ k \otimes \mathcal{I}(k\varphi )\right)\right\} \\ & \leq \varlimsup _{k \to \infty } \frac{m!}{k^ m} \dim _{\mathbb C}\left\{ s|_ Y: s\in \mathrm{H}^0\left(X,T\otimes L^ k \otimes \mathcal{I}(k\varphi )\right)\right\} \\ & \leq \lim _{k \to \infty } \frac{m!}{k^ m} h^0\left(Y ,T|_ Y\otimes L|_ Y^ k\otimes \operatorname{Res}_ Y\mathcal{I}(k\varphi )\right)\\ & = \int _ Y \left(\theta |_ Y+\mathrm{dd}^{\mathrm{c}}\operatorname{Tr}_ Y^{\theta }(\varphi )\right)^ m. \end{aligned} \]

This completes the proof. $\square$

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