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19.5.23 The general case101012300A5F

Definition 19.5.24.75695A1B0FB3 Let $T\in \mathcal{Z}_+(X,\alpha )$. Take a Kähler form $\omega $ on $X$. We define

C022629C737B 19.5.25
\begin{equation} \Delta _{Y_{\bullet }}(T)=\bigcap _{j=1}^{\infty }\Delta _{Y_{\bullet }}(T+j^{-1}\omega ). \end{equation}

The same definition makes sense when $\alpha $ is only pseudo-effective.

This definition is clearly independent of the choice of $\omega $ by 19.5.20. Moreover, it extends definition 19.5.15 and definition 19.5.7 as a result of 19.5.20. The main properties of $\Delta _{Y_{\bullet }}(T)$ are summarized as follows:

Theorem 19.5.26.944BC14A8246 The convex bodies $\Delta _{Y_{\bullet }}(T)$ satisfy the following properties:

  1. 15D38B2026CE

    Suppose that $T\in \mathcal{Z}_+(X,\alpha )_{>0}$. We have

    E0BB6985CB53 19.5.27
    \begin{equation} \operatorname{vol}\Delta _{Y_{\bullet }}(T)=\frac{1}{n!}\operatorname{vol}T. \end{equation}
  2. 802F287FEE19

    For $T,S \in \mathcal{Z}_+(X,\alpha )$ satisfying $T\preceq _{\mathcal{I}}S$, we have

    \[ \Delta _{Y_{\bullet }}(T)\subseteq \Delta _{Y_{\bullet }}(S)\subseteq \Delta _{Y_{\bullet }}(\alpha ). \]
  3. 50506C1C2B02

    For any current $T\in \mathcal{Z}_+(X,\alpha )$ with minimal singularities, we have

    \[ \Delta _{Y_{\bullet }}(T)=\Delta _{Y_{\bullet }}(\alpha ). \]
  4. 9D03BAD80E42

    The map $\mathcal{Z}_+(X,\alpha )_{>0}\rightarrow \mathcal{K}_ n$ given by $T\mapsto \Delta _{Y_{\bullet }}(T)$ is continuous, where we endow the $d_ S$-pseudometric on $\mathcal{Z}_+(X,\alpha )_{>0}$ and the Hausdorff topology on $\mathcal{K}_ n$.

  5. D4C7EBFD1762

    Let $\pi \colon Y\rightarrow X$ be a proper bimeromorphic morphism, where $Y$ is a Kähler manifold. Assume that the lifting $(W_{\bullet },g)$ of $Y_{\bullet }$ to $Y$ exists. Then for any $T\in \mathcal{Z}_+(X,\alpha )_{>0}$, we have

    \[ \Delta _{W_{\bullet }}(\pi ^*T)=\Delta _{Y_{\bullet }}(T)g. \]
  6. 8185D18AFE2D

    For $T,S\in \mathcal{Z}_+(X,\alpha )$, we have

    0696155A37C7 19.5.28
    \begin{equation} \Delta _{Y_{\bullet }}(T)+\Delta _{Y_{\bullet }}(S)\subseteq \Delta _{Y_{\bullet }}(T+S). \end{equation}

Proof. 1 By (19.5.25) and (19.5.19), for any Kähler form $\omega $ on $X$,

\[ \operatorname{vol}\Delta _{Y_{\bullet }}(T)=\lim _{j\to \infty }\operatorname{vol}\Delta _{Y_{\bullet }}(T+j^{-1}\omega )=\frac{1}{n!}\lim _{j\to \infty }\operatorname{vol}(T+j^{-1}\omega ). \]

The right-hand side is computed in proposition 18.1.27. Hence, (19.5.27) follows.

2 Fix a Kähler form $\omega $ on $X$. By corollary 19.5.17, for each $j\geq 1$,

\[ \Delta _{Y_{\bullet }}(T+j^{-1}\omega )\subseteq \Delta _{Y_{\bullet }}(S+j^{-1}\omega )\subseteq \Delta _{Y_{\bullet }}(\alpha +j^{-1}\{ \omega \} ). \]

It remains to show that

\[ \Delta _{Y_{\bullet }}(\alpha )=\bigcap _{j=1}^{\infty }\Delta _{Y_{\bullet }}(\alpha +j^{-1}\{ \omega \} ). \]

The $\subseteq $ direction is clear. Comparing the volumes using theorem 19.5.4, we conclude that equality holds.

3 This follows from 1 and 2.

4 Let $(T_ j)_ j$ be a sequence in $\mathcal{Z}_+(X,\alpha )_{>0}$ converging to $T\in \mathcal{Z}_+(X,\alpha )_{>0}$ with respect to $d_ S$. We want to show that $\Delta _{Y_{\bullet }}(T_ j)\xrightarrow {d_{\mathrm{Haus}}} \Delta _{Y_{\bullet }}(T)$. By 2, replacing the potential of each current by its $\mathcal I$-model representative does not change its partial Okounkov body. Thus we may represent the singularity types by model potentials of uniformly positive mass. Applying proposition 16.9.17, after passing to a subsequence we may sandwich the sequence between an increasing and a decreasing sequence converging to $T$ in $d_ S$. Hence it suffices to treat the monotone cases.

In either monotone case, 2 gives monotonicity of the convex bodies, while 1 and corollary 16.9.13 give convergence of their volumes to $\operatorname{vol}\Delta _{Y_{\bullet }}(T)$. Every Hausdorff cluster point is therefore a monotone limit with the same volume as $\Delta _{Y_{\bullet }}(T)$, hence is equal to $\Delta _{Y_{\bullet }}(T)$.

5 We may assume that $T$ is $\mathcal{I}$-good. It follows from 4 and theorem 18.1.2 that we may reduce to the case where $T$ has analytic singularities. Our assertion follows from 19.5.11.

6 By (19.5.25), in order to prove (19.5.28), we may assume that $T$ and $S$ are both Kähler currents. Take quasi-equisingular approximations $(T_ j)_ j$ and $(S_ j)_ j$ of $T$ and $S$ respectively. By theorem 16.9.45, $T_ j+S_ j\xrightarrow {d_ S} T+S$. By 4, we may therefore assume that $T$ and $S$ have analytic singularities. Replacing $X$ by a suitable modification, we may assume that $T$ and $S$ both have log singularities, say

\[ T=[D]+R,\quad S=[D']+R', \]

where $D$ and $D'$ are $\mathbb {Q}$-divisors on $X$ and $R$ and $R'$ are closed positive $(1,1)$-currents with bounded potentials. We need to show that

\[ \Delta _{Y_{\bullet }}(\{ R\} )+\Delta _{Y_{\bullet }}(\{ R'\} )+\nu _{Y_{\bullet }}([D])+\nu _{Y_{\bullet }}([D'])\subseteq \Delta _{Y_{\bullet }}(\{ R+R'\} )+\nu _{Y_{\bullet }}([D+D']). \]

By proposition 19.3.9, this is equivalent to

\[ \Delta _{Y_{\bullet }}(\{ R\} )+\Delta _{Y_{\bullet }}(\{ R'\} )\subseteq \Delta _{Y_{\bullet }}(\{ R+R'\} ), \]

which is already proved in theorem 19.5.4. $\square$

Corollary 19.5.29.68273CCE4042 Assume that $L$ is a big line bundle on $X$ and $h$ is a plurisubharmonic metric on $L$ with positive volume. Then

46A0E609C71E 19.5.30
\begin{equation} \Delta _{Y_{\bullet }}(\mathrm{dd}^{\mathrm{c}}h)=\Delta _{Y_{\bullet }}(L,h). \end{equation}
Similarly, the definition (19.4.38) is compatible with the definition in definition 19.5.24.

Proof. We may assume that $\mathrm{dd}^{\mathrm{c}}h$ has positive mass and is $\mathcal{I}$-good. By the $d_ S$-continuity of both sides of (19.5.30) as proved in theorem 19.5.26 and theorem 19.4.30, together with theorem 18.1.2, we may assume that $\mathrm{dd}^{\mathrm{c}}h$ has analytic singularities.

In this case, using the birational invariance of both sides of (19.5.30) as proved in proposition 19.4.32 and theorem 19.5.26, we may assume that $\mathrm{dd}^{\mathrm{c}}h$ has log singularities. Finally, after all these reductions, the equality (19.5.30) holds by construction. $\square$

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