18.1.44 The volumes of Hermitian pseudo-effective line bundles91B566E4374E

Proof. Step 1. Reduce to the case of log singularities.

Let $\pi \colon Y\rightarrow X$ be a log resolution of $\varphi $. In this case, for each $k\in \mathbb {Z}_{>0}$, we have

By proposition 16.6.10, we have

Therefore, it suffices to argue (18.1.47) with $K_{Y/X}\otimes \pi ^*T$, $\pi ^*L$ and $\pi ^*h$ in place of $T$, $L$ and $h$.

Note that $\theta _{\varphi }$ is no longer a Kähler current, but it still has positive mass.

Step 2. Assume that $\varphi $ has log singularities along an effective $\mathbb {Q}$-divisor $D$. We may assume that $D$ is simple normal crossing by replacing $X$ by a modification and $D$ by the pull-back. We decompose $D$ into irreducible components, say

In this case, we can easily compute

for each $k\in \mathbb {Z}_{>0}$. Observe that $L-D$ is nef (see 5.18.5). Choose an integer $q>0$ so that $qD$ is integral. For each residue class $s=0,\ldots ,q-1$, we have

Hence, on the residue class $k=mq+s$, the line bundle

is a fixed twist of $\left(q(L-D)\right)^ m$. Applying the asymptotic Riemann–Roch theorem [Corollary 1.4.41, PAG1] to the nef line bundle $q(L-D)$ and then letting $m\to \infty $ in each residue class, we conclude that

Observe that by proposition 5.20.4,

where $S$ is a closed positive $(1,1)$-current with bounded potential. Therefore,

By example 18.1.10, we know that the right-hand side is exactly $\operatorname{vol}\theta _{\varphi }$. $\square$

Proof of theorem 18.1.46. Step 1. We first handle the case where $\theta _{\varphi }$ is a Kähler current. Fix a Kähler form $\omega \geq \theta $ on $X$ such that $\theta _{\varphi }\geq 2\delta \omega $ for some $\delta \in (0,1)$.

Let $(\varphi _ j)_ j$ be a quasi-equisingular approximation of $\varphi $ in $\mathrm{PSH}(X,\theta )$. We may assume that $\theta _{\varphi _ j}\geq \delta \omega $ for all $j$. From proposition 18.1.48, we know that for each $j\geq 1$,

It follows from theorem 18.1.2 and theorem 16.9.57 that the right-hand side converges to $\operatorname{vol}\theta _{\varphi }$ as $j\to \infty $. Therefore,

Conversely, fix an integer $N>\delta ^{-1}$. From theorem 18.1.2 and theorem 16.9.30, we know that

Therefore, by 13.13.30, we can find $j_0>0$ such that for $j\geq j_0$, there is $\psi \in \mathrm{PSH}(X,\theta )_{>0}$ (depending on $j$) with

For each $k>0$, we write $k=k'N-r$, where $k'\in \mathbb {N}$ and $r\in \{ 0,1,\ldots ,N-1\} $. Then we compute for $j\geq j_0$ and large enough $k$ (to be specified shortly) that

where the third line follows from (18.1.50), the fourth line can be argued as follows: For large enough $k$, there is a non-zero section $s\in \mathrm{H}^0(X,L^{k'}\otimes \mathcal{I}(k'\psi ))$ by 13.13.48. It follows from 5.18.15 that for large enough $k$,

It follows that multiplication by $s$ gives an injective map

Next observe that

Since $N>\delta ^{-1}$, this is a Kähler current. Hence proposition 18.1.48 is applicable. We let $k\to \infty $ to conclude that

where $C$ is a constant independent of $N$ and $j$. Letting $j\to \infty $ and then $N\to \infty $ and using (18.1.49), we find that

Therefore, (18.1.47) follows.

Step 2. We handle the case where $\operatorname{vol}\theta _{\varphi }>0$. We may assume that $\varphi $ is $\mathcal{I}$-model.

First observe that $L$ is big by proposition 13.13.3. Hence $X$ is projective. Take a very ample line bundle $A$ on $X$ and a Kähler form $\omega $ in $c_1(A)$. Take a Hermitian metric $h_ A$ on $A$ with $\mathrm{dd}^{\mathrm{c}}h_ A=\omega $.

Fix $N\in \mathbb {Z}_{>0}$. We decompose any $k>0$ as $k=k'N+r$ with $k'\in \mathbb {N}$ and $r\in \{ 0,1,\ldots ,N-1\} $. Then

Therefore,

where the third line uses multiplication by a non-zero section of $A^{k'}$, and we have applied Step 1 to the Hermitian pseudo-effective line bundle $(L^ N\otimes A,h^{\otimes N}\otimes h_ A)$ on the fourth line. On the other hand, since $\varphi $ is $\mathcal{I}$-good by example 18.1.11, 18.1.8 gives

It follows from proposition 16.3.10 that

Letting $N\to \infty $, we conclude

It remains to argue the reverse inequality.

Choose $\psi \in \mathrm{PSH}(X,\theta )$ such that $\theta _{\psi }$ is a Kähler current and $\psi \leq \varphi $. The existence of $\psi $ is guaranteed by 13.13.47. Then for any $t\in (0,1)$, we set

It follows again from Step 1 that

On the other hand, by theorem 18.1.32 and corollary 16.9.38,

So we find

We conclude (18.1.47) in this case.

Step 3. We finally handle the case where $\operatorname{vol}\theta _{\varphi }=0$. Replacing $\varphi $ by $P_{\theta }[\varphi ]_{\mathcal{I}}$, we may assume that $\varphi $ is $\mathcal{I}$-model.

Assume that (18.1.47) fails. That is,

then $L$ is a big line bundle and hence $X$ is projective.

Fix a very ample line bundle $A$ on $X$ and a Kähler form $\omega \in c_1(A)$. Take a decreasing sequence $(\epsilon _ j)_ j$ of rational numbers with limit $0$ and a quasi-equisingular approximation $(\varphi _ j)_ j$ of $\varphi $ with $\varphi _ j\in \mathrm{PSH}(X,\theta +\epsilon _ j\omega )_{>0}$.

We claim that as $j\to \infty $, the sequence $P_{\theta +\epsilon _ j\omega }[\varphi _ j]$ is decreasing with limit $\varphi $.

It is clear that this sequence is decreasing. Let $\psi $ denote its limit for the moment. It is also clear that $\psi \geq \varphi $. Since $\varphi $ is $\mathcal{I}$-model, it remains to show that $\psi \preceq _{\mathcal{I}}\varphi $. Fix $\lambda >0$. By strong openness, choose $\lambda '>\lambda $ such that $\mathcal{I}(\lambda \psi )=\mathcal{I}(\lambda '\psi )$. For $j$ large enough, the quasi-equisingularity of $(\varphi _ j)_ j$ gives

where the first inclusion uses $\psi \leq P_{\theta +\epsilon _ j\omega }[\varphi _ j]$ and $P_{\theta +\epsilon _ j\omega }[\varphi _ j]\sim _{\mathcal{I}}\varphi _ j$. Thus $\psi \preceq _{\mathcal{I}}\varphi $, and the claim follows.

By our claim, proposition 16.3.10 and proposition 16.4.4, we find that

Fix $j>0$, take an integer $N>0$ so that $N\epsilon _ j$ is an integer. Then we compute

where the third line follows by writing $k=Nk'+a$ as before, we applied Step 2 on the last line. Dividing by $N^ n$, we find that

Since we know (18.1.51), letting $j\to \infty $, we conclude that

which is a contradiction. Hence (18.1.47) is established in full generality. $\square$

Proof. It suffices to observe that

for all $k>0$, since for any holomorphic section $s$ of $L^ k$, the function $\log |s|_{h^ k}^2$ is $\theta $-psh thanks to proposition 5.20.4 and therefore, we have

In particular, $s\in \mathrm{H}^0\left(X,L^ k \otimes \mathcal{I}\left(kV_{\theta }\right) \right)$. $\square$

Proof. Thanks to 18.1.8, we may assume that $\theta \in c_1(L)$ for some toric invariant ample line bundle $L$. In this case, the result follows from theorem 18.1.2, theorem 18.1.46 and theorem 21.12.31. $\square$