19.4.50 Recover Lelong numbers from partial Okounkov bodies

Theorem 19.4.51.7CC4ADAC32DC Let $E$ be a prime divisor on $X$. Let $Y_{\bullet }$ be an admissible flag with $E=Y_1$. Then

CA1290F03811 19.4.52

\begin{equation} \nu (\varphi ,E)=\min _{x\in \Delta _{Y_{\bullet }}(\theta ,\varphi )}x_1. \end{equation}

The picture in 19.4.53 illustrates the formula: the left supporting hyperplane of $\Delta _{Y_{\bullet }}(\theta ,\varphi )$ is given by $x_1=\nu (\varphi ,Y_1)$.

Proof. Replacing $\varphi $ by $P_{\theta }[\varphi ]_{\mathcal{I}}$, we may assume that $\varphi $ is $\mathcal{I}$-good.

Step 1. We first reduce to the case where $\varphi $ has analytic singularities.

By theorem 18.1.2, we can find a sequence $(\varphi _ j)_ j$ in $\mathrm{PSH}(X,\theta )_{>0}$ with analytic singularities such that $\varphi _ j\xrightarrow {d_ S}\varphi $. It follows from theorem 19.4.30 that

\[ \Delta _{Y_{\bullet }}(\theta ,\varphi _ j)\xrightarrow {d_{\mathrm{Haus}}}\Delta _{Y_{\bullet }}(\theta ,\varphi ). \]

Therefore, by 19.7.10,

\[ \lim _{j\to \infty }\min _{x\in \Delta _{Y_{\bullet }}(\theta ,\varphi _ j)}x_1=\min _{x\in \Delta _{Y_{\bullet }}(\theta ,\varphi )}x_1. \]

In view of theorem 16.9.55, it suffices to prove (19.4.52) with $\varphi _ j$ in place of $\varphi $.

Step 2. Assume that $\varphi $ has analytic singularities. In view of proposition 19.4.32 and theorem 5.18.8, after replacing $X$ by a modification, we may assume that $\varphi $ has log singularities along an effective $\mathbb {Q}$-divisor $F$.

Perturbing $L$ by an ample $\mathbb {Q}$-line bundle by proposition 19.4.39, we may assume that $\theta _{\varphi }$ is a Kähler current. Therefore, $L-F$ is ample by 5.18.5. Finally, by rescaling, we may assume that $F$ is a divisor and $L$ is a line bundle.

By theorem 19.4.41, we know that

\[ \min _{x\in \Delta _{Y_{\bullet }}(\theta ,\varphi )}x_1=\adjustlimits \lim _{k\to \infty }\min _{x\in \Delta _ k(\theta ,\varphi )}x_1. \]

By definition,

\[ \min _{x\in \Delta _ k(\theta ,\varphi )}x_1=k^{-1}\operatorname{ord}_ E \mathrm{H}^0\left(X,L^ k\otimes \mathcal{I}(k\varphi )\right). \]

In view of proposition 5.16.15, it remains to show that

0450C5C72C83 19.4.54

\begin{equation} \lim _{k\to \infty }k^{-1}\operatorname{ord}_ E \mathrm{H}^0\left(X,L^ k\otimes \mathcal{I}(k\varphi )\right)=\lim _{k\to \infty }k^{-1}\operatorname{ord}_ E \mathcal{I}(k\varphi ). \end{equation}

The $\geq $ direction is trivial, we prove the converse. Observe that

\[ \mathrm{H}^0\left(X,L^ k\otimes \mathcal{I}(k\varphi )\right)=\mathrm{H}^0\left(X,L^ k\otimes \mathcal{O}_ X(-kF)\right),\quad \mathcal{I}(k\varphi )=\mathcal{O}(-kF). \]

As $L-F$ is ample, for large enough $k$, the line bundle $k(L-F)$ is globally generated. Hence we can choose a section of $k(L-F)$ which does not vanish at the generic point of $E$. Viewed as a section of $L^ k\otimes \mathcal O_ X(-kF)$, it has vanishing order exactly $\operatorname{ord}_ E(kF)$ along $E$. Therefore,

\[ \operatorname{ord}_ E \mathrm{H}^0\left(X,L^ k\otimes \mathcal{O}_ X(-kF)\right)=\operatorname{ord}_ E(kF). \]

Thus, (19.4.54) follows. $\square$

Corollary 19.4.55.C3CA9BD5D216 Let $\varphi ,\psi \in \mathrm{PSH}(X,\theta )_{>0}$. If

\[ \Delta _{W_{\bullet }}(\pi ^*\theta ,\pi ^*\varphi )\subseteq \Delta _{W_{\bullet }}(\pi ^*\theta ,\pi ^*\psi ) \]

for all modifications $\pi \colon Y\rightarrow X$ and all admissible flags $W_{\bullet }$ on $Y$, then $\varphi \preceq _{\mathcal{I}} \psi $.

Proof. By theorem 19.4.51, the assumed inclusions imply that for every prime divisor $E$ on every modification $\pi $,

\[ \nu (\pi ^*\varphi ,E)\geq \nu (\pi ^*\psi ,E). \]

Hence $\varphi \preceq _{\mathcal{I}}\psi $ by proposition 16.8.9. $\square$

Corollary 19.4.56.C08CC234CAB9 Let $E$ be a prime divisor over $X$. Then

3C67C9063373 19.4.57

\begin{equation} \nu (V_{\theta },E)=\lim _{k\to \infty }\frac{1}{k}\operatorname{ord}_ E \mathrm{H}^0(X,L^ k). \end{equation}

Proof. After replacing $X$ by a modification, we may assume that $E$ is a prime divisor on $X$. Note that since $X$ is projective, we can always find an admissible flag with $E$ being the first component.

Then (19.4.57) follows from theorem 19.4.51 and the fact that $\Delta _{Y_{\bullet }}(\theta ,V_{\theta })=\Delta _{Y_{\bullet }}(L)$ for any admissible flag $Y_{\bullet }$ on $X$. $\square$

19.4.50 Recover Lelong numbers from partial Okounkov bodies`6EC7E7A8EF44`

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19.4.50 Recover Lelong numbers from partial Okounkov bodies6EC7E7A8EF44

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19.4.50 Recover Lelong numbers from partial Okounkov bodies`6EC7E7A8EF44`