PomeCorollary 21.12.35.05A4124B7E31 Let $\varphi \in \mathrm{PSH}_{\mathrm{tor}}(X,\omega )$. Then
\[ \lim _{k\to \infty }\frac{n!}{k^ n}h^0\left(X,L^ k\otimes \mathcal{I}(k\varphi )\right)= n!\operatorname{vol}\Delta (\omega ,\varphi ). \]
Corollary 21.12.3505A4124B7E31
Proof. Take $k>0$. Let $F_{0,k}$ denote the Guillemin potential associated with the polytope $kP$. On $T(\mathbb C)$, set
\[ \widetilde\varphi _ k=\operatorname{Trop}^*(kF_\varphi -F_{0,k}). \]
Then $\widetilde\varphi _ k$ is the toric potential, with respect to the reference metric associated with $kP$, corresponding to the singular metric $h^ k\exp (-k\varphi )$ on $L^ k$. It differs from $k\varphi $ by the smooth toric function $\operatorname{Trop}^*(kF_0-F_{0,k})$. Hence
\[ \mathcal I(\widetilde\varphi _ k)=\mathcal I(k\varphi ). \]
Moreover, the convex function associated with $\widetilde\varphi _ k$ is $kF_\varphi $, so its Legendre transform is
\[ (kF_\varphi )^*(m)=kG_\varphi (m/k). \]
Thus the Newton body of $\widetilde\varphi _ k$ is $k\Delta (\omega ,\varphi )$.
Applying theorem 21.12.31 1 to the line bundle $L^ k$ and the potential $k\varphi $, whose Newton body is $k\Delta (\omega ,\varphi )$, we have
\[ \# \left( k\Delta (\omega ,\varphi )\cap M\right)\leq h^0\left(X,L^ k\otimes \mathcal{I}(k\varphi )\right). \]
Here $\# $ denotes the number of elements. Therefore,
\[ \lim _{k\to \infty }k^{-n}\# \left( k\Delta (\omega ,\varphi )\cap M\right)\leq \varliminf _{k\to \infty } k^{-n} h^0\left(X,L^ k\otimes \mathcal{I}(k\varphi )\right). \]
The left-hand side is just $\operatorname{vol}\Delta (\omega ,\varphi )$, as follows from theorem 19.7.20. So we conclude that
\[ \varliminf _{k\to \infty }\frac{n!}{k^ n}h^0\left(X,L^ k\otimes \mathcal{I}(k\varphi )\right)\geq n!\operatorname{vol}\Delta (\omega ,\varphi ). \]
It remains to prove the reverse inequality.
We choose an identification $M\cong \mathbb {Z}^ n$. In particular, there is a natural distance $\mathrm{d}$ on $M_{\mathbb {R}}\cong \mathbb {R}^ n$. Given a convex body $K$ and $\epsilon >0$, we let
\[ K^{\epsilon }\coloneqq \left\{ x\in \mathbb {R}^ n: \mathrm{d}(x,K)\leq \epsilon \right\} . \]
This is again a convex body, since it can be realized as the Minkowski sum of $K$ with a closed ball of radius $\epsilon $.
Applying theorem 21.12.31 2 to the polytope $kP$ and the potential $\widetilde\varphi _ k$, we can find $C>0$ uniform in $k$ such that for each $k\in \mathbb {Z}_{>0}$,
\[ \bigl \{ m\in M:\chi ^ m\in \mathrm{H}^0(X,L^ k\otimes \mathcal{I}(k\varphi ))\bigr \} \subseteq \left(k\Delta (\omega ,\varphi )\right)^ C. \]
Indeed, if a point $m\in kP\cap M$ is at distance at least $C$ from $k\Delta (\omega ,\varphi )$, the separating hyperplane theorem gives a unit vector $v\in N_{\mathbb R}$ such that
\[ \langle m,v\rangle -\operatorname{Supp}_{k\Delta (\omega ,\varphi )}(v)\geq C. \]
Since the fan $\Sigma $ is finite and covers $N_{\mathbb R}$, there is a constant $\Lambda >0$ with the following property: every unit vector $v\in N_{\mathbb R}$ can be written in the form
\[ v=-\lambda \sum _ i c_ i u_{\rho _ i},\quad 0<\lambda \leq \Lambda ,\quad c_ i>0,\quad \sum _ i c_ i=1, \]
where the rays $\rho _ i$ are contained in a single cone of $\Sigma $. Put
\[ u\coloneqq \sum _ i c_ i u_{\rho _ i}. \]
Since the support function is positively homogeneous, we have
\[ \begin{aligned} \langle m,-u\rangle -\operatorname{Supp}_{k\Delta (\omega ,\varphi )}(-u)& =\lambda ^{-1}\left(\langle m,v\rangle -\operatorname{Supp}_{k\Delta (\omega ,\varphi )}(v)\right)\\ & \geq C/\Lambda . \end{aligned} \]
Choosing $C>\Lambda $ makes (21.12.32) applicable, uniformly in $k$.
Fix $\epsilon >0$. Then for large enough $k$, we have
\[ \left(k\Delta (\omega ,\varphi )\right)^{C}\subseteq k\left(\Delta (\omega ,\varphi )^{\epsilon }\right). \]
It follows that
\[ h^0\left(X,L^ k\otimes \mathcal{I}(k\varphi )\right)\leq \# \left( k\left(\Delta (\omega ,\varphi )^{\epsilon }\right) \cap M\right). \]
Applying theorem 19.7.20 again, we conclude that
\[ \varlimsup _{k\to \infty } \frac{n!}{k^ n} h^0\left(X,L^ k\otimes \mathcal{I}(k\varphi )\right)\leq n! \operatorname{vol}\left(\Delta (\omega ,\varphi )^{\epsilon }\right). \]
Letting $\epsilon \to 0+$ and applying theorem 19.7.5, we find
\[ \varlimsup _{k\to \infty } \frac{n!}{k^ n} h^0\left(X,L^ k\otimes \mathcal{I}(k\varphi )\right)\leq n! \operatorname{vol}\Delta (\omega ,\varphi ). \]
Our assertion follows. $\square$
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