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22.1.54 Duistermaat–Heckman measures3C0D12E80020

Let $X$ be a connected compact Kähler manifold of dimension $n$ and $\theta $ be a closed real smooth $(1,1)$-form on $X$ representing a big cohomology class.

Definition 22.1.55.9ADEFD2FF381 Assume that $X$ admits a smooth flag $Y_{\bullet }$. Let $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$. The Duistermaat–Heckman measure $\operatorname{DH}(\Gamma )$ of $\Gamma $ is defined as

\[ \operatorname{DH}(\Gamma )\coloneqq n!\cdot \operatorname{DH}\left(\Delta _{Y_{\bullet }}(\theta ,\Gamma )\right). \]
Recall that $\Delta _{Y_{\bullet }}(\theta ,\Gamma )\in \mathrm{TC}(\Delta _{Y_{\bullet }}(\theta ,\Gamma _{-\infty }))$ is the Okounkov test curve defined in theorem 19.6.24. See definition 19.6.18 for the definition of the Duistermaat–Heckman measure of an Okounkov test curve.

Theorem 22.1.56.0AD7F80A793D Assume that $X$ admits a smooth flag $Y_{\bullet }$. The Duistermaat–Heckman measure $\operatorname{DH}(\Gamma )$ of $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$ in definition 22.1.55 is independent of the choice of the smooth flag $Y_{\bullet }$. Furthermore, for any $m\in \mathbb {Z}_{>0}$, the following moment identity holds:

A51020D58E6F 22.1.57
\begin{equation} \int _{\mathbb {R}}x^ m\operatorname{DH}(\Gamma )(x)=\Gamma _{\max }^ m\operatorname{vol}\Gamma +m\int _{-\infty }^{\Gamma _{\max }}\tau ^{m-1}\Bigl ( \operatorname{vol}(\theta +\mathrm{dd}^{\mathrm{c}}\Gamma _{\tau })-\operatorname{vol}\Gamma \Bigr )\, \mathrm{d}\tau , \end{equation}

and

51031202809C 22.1.58
\begin{equation} \int _{\mathbb {R}}\operatorname{DH}(\Gamma )=\operatorname{vol}\Gamma . \end{equation}
For each $m\in \mathbb {N}$, $\int _{\mathbb {R}}x^ m\operatorname{DH}(\Gamma )(x)$ is indeed $n!$ times the $m$-th moment of the random variable $G[\Delta _{Y_{\bullet }}(\theta ,\Gamma )]$, by definition 22.1.55, definition 19.6.18.

Proof. We observe that (22.1.57) holds by the computations in proposition 19.6.19 and theorem 19.5.26 1. It is independent of the choice of the flag. Furthermore, (22.1.58) follows immediately from theorem 19.5.26 1.

Assume first that $\Gamma $ is bounded. Since the Duistermaat–Heckman measure has bounded support in this case (cf. theorem 19.6.13), we conclude that $\operatorname{DH}(\Gamma )$ is uniquely determined.

By translating the parameter, it suffices to treat the case $\Gamma _{\max }=0$. Indeed, if $a=\Gamma _{\max }$ and

\[ \widetilde\Gamma _\tau \coloneqq \Gamma _{\tau +a}, \]

then $\widetilde\Gamma _{\max }=0$, and the associated test function is translated by $a$. Consequently,

\[ \operatorname{DH}(\Gamma )=(x\mapsto x+a)_*\operatorname{DH}(\widetilde\Gamma ). \]

Thus independence of the flag and the moment identity for $\widetilde\Gamma $ imply the corresponding assertions for $\Gamma $.

For each $\epsilon >0$, we define $\Gamma ^{\epsilon }\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$ as follows:

  1. 36B347B9186A

    Let $\Gamma ^{\epsilon }_{\max }=0$, and

  2. 12A1009F3470

    we set

    \[ \Gamma ^{\epsilon }_{\tau }= \begin{cases} \Gamma _{-\infty },& \text{ if }\tau \leq -\epsilon ^{-1},\\ P_{\theta }\left[ (1+\epsilon \tau )\Gamma _{\tau }-\epsilon \tau \Gamma _{-\infty } \right],& \text{ if }\tau \in \left(-\epsilon ^{-1},0\right). \end{cases} \]

Let us check that $\Gamma ^\epsilon $ is indeed an element of $\mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$. The concavity and monotonicity in $\tau $ follow from proposition 16.4.2. Since the $\Gamma _\tau $'s are $\mathcal I$-model, they are $\mathcal I$-good by example 18.1.11; hence

\[ (1+\epsilon \tau )\Gamma _\tau -\epsilon \tau \Gamma _{-\infty } \]

is $\mathcal I$-good by proposition 18.1.17. Therefore its $P$-envelope is also its $\mathcal I$-envelope, and the corresponding level is $\mathcal I$-model. The value at $-\infty $ is $\Gamma _{-\infty }$. Thus $\Gamma ^\epsilon \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$.

For each fixed $\tau <0$, the argument of theorem 17.13.21 Step 3.3 gives

\[ \Gamma ^\epsilon _\tau \searrow \Gamma _\tau \quad \textup{as }\epsilon \to 0+. \]

By proposition 16.4.4 and corollary 16.9.14, this convergence is in $d_ S$. Hence theorem 19.5.26 4 gives

\[ \Delta _{Y_\bullet }(\theta ,\Gamma ^\epsilon _\tau )\xrightarrow {d_{\mathrm{Haus}}}\Delta _{Y_\bullet }(\theta ,\Gamma _\tau ) \]

for every $\tau <0$. Since

\[ \Gamma ^\epsilon _{-\infty }=\Gamma _{-\infty } \]

for all $\epsilon $, the underlying convex body is fixed. Therefore 19.6.22 applies and gives

\[ \operatorname{DH}(\Gamma ^\epsilon )\rightharpoonup \operatorname{DH}(\Gamma ). \]

It follows that $\operatorname{DH}(\Gamma )$ is independent of the choice of the flag. $\square$

More generally, when $X$ does not admit a smooth flag, we may choose a modification $\pi \colon Y\rightarrow X$ so that $Y$ admits a flag. We define

665B92171EA8 22.1.59
\begin{equation} \operatorname{DH}(\Gamma )\coloneqq \operatorname{DH}(\pi ^*\Gamma ). \end{equation}

It follows from theorem 19.5.26 5 that this measure is independent of the choice of $\pi $.

Proposition 22.1.60.492A145E9067 Let

\[ (\Gamma ^ i)_{i\geq 1}\subseteq \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0},\quad \Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}. \]

Assume one of the following conditions holds:

  1. 577BE71DB8C5

    The sequence $(\Gamma ^ i)_{i\geq 1}$ is decreasing and $\Gamma =\inf _{i\geq 1}\Gamma ^ i$. Assume that

    E10A583B2E40 22.1.61
    \begin{equation} \operatorname{vol}\Gamma =\lim _{i\to \infty }\operatorname{vol}\Gamma ^ i. \end{equation}
  2. 75E1B24CB49A

    The sequence $(\Gamma ^ i)_{i\geq 1}$ is increasing and $\Gamma =\operatorname *{sup}_{i\geq 1}{}\! {}^{*}\Gamma ^ i$.

Then

636AFE1086C1 22.1.62
\begin{equation} \operatorname{DH}(\Gamma ^ i)\rightharpoonup \operatorname{DH}(\Gamma ). \end{equation}

Proof. We may assume that $X$ admits a smooth flag $Y_{\bullet }$.

Assume 1. Let

\[ \eta \coloneqq \inf _{i\geq 1}\Gamma ^ i_{-\infty }. \]

Since $\Gamma _\tau \leq \Gamma ^ i_{-\infty }$ for every $i$ and every $\tau <\Gamma _{\max }$, we have

\[ \Gamma _{-\infty }\leq \eta . \]

By proposition 16.4.4, the decreasing limit $\eta $ is model and

\[ \int _ X\theta _\eta ^ n=\lim _{i\to \infty }\int _ X\theta _{\Gamma ^ i_{-\infty }}^ n=\lim _{i\to \infty }\operatorname{vol}\Gamma ^ i. \]

Using (22.1.61), this equals

\[ \operatorname{vol}\Gamma =\int _ X\theta _{\Gamma _{-\infty }}^ n. \]

Since $\Gamma _{-\infty }\leq \eta $ and both potentials are model, proposition 16.8.7 gives

\[ \Gamma _{-\infty }=\eta . \]

Thus

\[ \Gamma _{-\infty }=\inf _{i\geq 1}\Gamma ^ i_{-\infty }. \]

We want to derive (22.1.62) from 19.6.22. It boils down to prove the following: For any $\tau <\Gamma _{\max }$, we have

\[ \Delta _{Y_{\bullet }}(\theta ,\Gamma ^ i_{\tau })\xrightarrow {d_{\mathrm{Haus}}} \Delta _{Y_{\bullet }}(\theta ,\Gamma _{\tau }). \]

For fixed $\tau <\Gamma _{\max }$, the potentials $\Gamma ^ i_\tau $ decrease to $\Gamma _\tau $. By proposition 16.4.4, their masses converge to the mass of $\Gamma _\tau $; hence $\Gamma ^ i_\tau \xrightarrow {d_ S}\Gamma _\tau $ by corollary 16.9.14. Applying the continuity of the partial Okounkov body map, theorem 19.5.26 4, gives

\[ \Delta _{Y_{\bullet }}(\theta ,\Gamma ^ i_{\tau })\xrightarrow {d_{\mathrm{Haus}}}\Delta _{Y_{\bullet }}(\theta ,\Gamma _{\tau }). \]

The proof under the assumption 2 is similar. We only need to apply 19.6.23 instead of 19.6.22. $\square$

Remark 22.1.63.CF750A31909E When $\{ \theta \} $ is a Hodge class and $\Gamma $ is induced by a test configuration as in example 17.14.10 and 17.14.14, our Duistermaat–Heckman measure coincides with the more traditional definition of [Section 3.2, BHJ17]. This is explained in [Remark 7.17, Xia21].

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