Pome

21.13.24 Newton bodiesB09F60655D0A

Let $\mathrm{PSH}_{\mathrm{tor}}(X,\theta )$ be the set of $T_ c$-invariant functions in $\mathrm{PSH}(X,\theta )$.

Definition 21.13.25.D7E78CFDB281 A function $\varphi \in \mathrm{PSH}_{\mathrm{tor}}(X,\theta )$ can be written as

\[ \varphi |_{T(\mathbb {C})}=\operatorname{Trop}^*f \]

for some unique function $f\colon N_{\mathbb {R}}\rightarrow [-\infty ,\infty )$. Then we define $F_{\varphi }\colon N_{\mathbb {R}}\rightarrow \mathbb {R}$ as follows:

E09FCEF2675C 21.13.26
\begin{equation} F_{\varphi }=F_{\theta }+f. \end{equation}
Observe that $F_{\varphi }$ is a convex function and takes finite values by 21.12.7. In particular, $f$ is also real-valued. Once $D$ and $h$ are fixed, $F_{\varphi }$ is well-defined up to a constant since $F_{\theta }$ is.

Definition 21.13.27.22C597E4DEB2 Let $\varphi \in \mathrm{PSH}_{\mathrm{tor}}(X,\theta )$, we define its Newton body as

\[ \Delta (\theta ,\varphi )\coloneqq \overline{\nabla F_{\varphi }(N_{\mathbb {R}})}\subseteq M_{\mathbb {R}}. \]
Here the gradient is understood in the sense of Alexandrov, see definition 21.11.30 for the details. Note that $\Delta (\theta ,\varphi )$ is independent of the choice of $s_ D$.

The Newton body $\Delta (\theta ,\varphi )$ depends on the choice of $D$, not only on the associated line bundle $L$: A different choice of $D$ inducing the same line bundle corresponds to a translation of $\Delta (\theta ,\varphi )$. We will see in a while (theorem 21.13.39) that once $D$ is fixed $\Delta (\theta ,\varphi )$ depends only on the current $\theta _{\varphi }$. Hence, the choice of $h$ is irrelevant.

For the moment, it is not clear if $\Delta (\theta ,\varphi )\subseteq P_ D$ or not. We shall prove this result using the theory of partial Okounkov bodies in the next section.

Proposition 21.13.28.49A9A66B3A4D Let $\varphi \in \mathrm{PSH}_{\mathrm{tor}}(X,\theta )$. Then

04F5C34B36CA 21.13.29
\begin{equation} \operatorname{Trop}_* \left(\theta |_{T(\mathbb {C})}+\mathrm{dd}^{\mathrm{c}}\varphi |_{T(\mathbb {C})} \right)^ n=\operatorname{MA}_{\mathbb {R}}\left(F_{\varphi }\right). \end{equation}

In particular,

5FCC3FDF4C75 21.13.30
\begin{equation} \int _ X \theta _{\varphi }^ n =n! \operatorname{vol}\Delta (\theta ,\varphi ) \end{equation}

Proof. Let $F_0$ be a smooth convex function on $N_{\mathbb {R}}$ such that $\mathrm{dd}^{\mathrm{c}}\operatorname{Trop}^* F_0$ can be extended to a Kähler form on $X$. For example, Guillemin's construction (21.12.9) with respect to a suitable Delzant polytope gives such an example.

Write $\omega $ for a Kähler form on $X$ extending $\mathrm{dd}^{\mathrm{c}}\operatorname{Trop}^*F_0$. Then for any large enough $C>0$, $\theta +C\omega $ is a Kähler form. So we conclude from proposition 21.12.24 that

\[ \operatorname{Trop}_* \left((\theta +C\omega )|_{T(\mathbb {C})}+\mathrm{dd}^{\mathrm{c}}\varphi |_{T(\mathbb {C})} \right)^ n=\operatorname{MA}_{\mathbb {R}}(F_{\varphi }+CF_0). \]

By the multi-linearity of the non-pluripolar product proposition 13.11.4 6, the left-hand side is a polynomial in $C$. By the expansion formula for the mixed real Monge–Ampère measure theorem 21.11.58 1, the right-hand side is also a polynomial in $C$. We conclude that the same holds for $C=0$. Therefore, (21.13.29) follows.

(21.13.30) is a direct consequence of (21.13.29). $\square$

Cite this page

Comments

Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.

Read this page’s discussion on GitHubPost a comment on GitHub