Pome

Theorem 21.13.451293B87EF09F

Theorem 21.13.45.1293B87EF09F We have

188F0AD7B28C 21.13.46
\begin{equation} F_{V_{\theta }}\in \mathcal{E}^{\infty }(N_{\mathbb {R}},P_ D).{}^{\href{#sgpt-a0000102047}{3}}\end{equation}

In particular,

82B62A741230 21.13.47
\begin{equation} \int _ X \theta _{V_{\theta }}^ n=n!\operatorname{vol}P_ D. \end{equation}

Proof. Taking $\varphi =V_{\theta }$ in theorem 21.13.39, we find

\[ \Phi _{\mathbb {R}}\left( \Delta (\theta ,V_{\theta })\right)=\Delta _{Y_{\bullet }}(\theta ,V_{\theta })=\Delta _{Y_{\bullet }}(L)=\Phi _{\mathbb {R}}(P_ D), \]

where we applied proposition 21.13.33 in the last equality. Therefore, (21.13.49) follows. In particular, (21.13.47) follows from proposition 21.13.28.

Next we prove (21.13.48). Take a smooth function $\chi \colon N_{\mathbb {R}}\rightarrow [0,1]$ so that

  1. 37909700D0C1

    $\operatorname{Supp}\chi $ is compact;

  2. C1605365568C

    $\chi \equiv 1$ on a neighbourhood of $0$.

Fix a $T_ c$-invariant volume form $\mu $ on $T(\mathbb {C})$. Then $(\operatorname{Trop}^*\chi )\mu $ can be regarded as a $T_ c$-invariant smooth non-negative measure on $X$. After choosing a suitable normalizing constant $C>0$, there is a unique solution

F3EAEAF94235 21.13.50
\begin{equation} \theta _{\varphi }^ n=C\left(\operatorname{Trop}^*\chi \right)\mu ,\quad \sup _ X \varphi =0,\quad \varphi \in \mathcal{E}^{\infty }(X,\theta ). \end{equation}

This is proved in [Theorem B, BEGZ10]. The uniqueness of $\varphi $ further guarantees its $T_ c$-invariance. Since $\varphi $ has minimal singularities, $F_{\varphi }\sim F_{V_{\theta }}$. Next, using proposition 21.13.28, the Monge–Ampère equation (21.13.50) implies

\[ \operatorname{MA}_{\mathbb {R}}\left(F_{\varphi }\right)=C \chi \operatorname{Trop}_*\mu . \]

It follows from the regularity theorem [Theorem 2.19, BB13] and (21.13.49) that $F_{\varphi }\sim \operatorname{Supp}_{P_ D}$. Hence, (21.13.48) follows. $\square$

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