Suppose we start with some symplectic potential $u$ and corresponding Kahler potential $\phi$. Then $\phi$ can be regarded as a Hermitian metric on the line bundle $L$ over the toric variety. Thus we have a natural $L^{2}$-metric on $H^{0}(X;L)$

$$\|s\|^{2}=\int_{X}|s|^{2}d\mu_{\phi},$$

where the pointwise norm $|s|$ is defined by $\phi$ and $d\mu_{\phi}$ is the volume form of the Kahler metric. Thus, starting with $u$ we get a collection of numbers $a_{\nu}=\|s_{\nu}\|^{-1}$. Now replace $L$ by $L^{k}$, as above. The same symplectic potential $u$ defines a metric on $L^{k}$ and we get a collection of numbers $a_{\nu}^{(k)}$ say, for $\nu\in\overline{P}\cap k^{-1}{\bf Z}^{n}$. One precise statement expressing the general idea above is that for each $\epsilon>0$ and compact subset $K\subset P$ there is a $k_{0}$ such that

$$|u(\nu)-k^{-1}\log a_{\nu}^{(k)}|<\epsilon,$$

once $k\geq k_{0}$, for all $\nu\in K\cap k^{-1}{\bf Z}^{n}$.

The proof of this is very simple. Go back to the case $k=1$ for the moment. Unravelling the definitions, the coefficients $a_{\nu}$ are given by

$$a_{\nu}^{-1}=\int_{{\bf R}^{n}}e^{-\phi}e^{\underline{t}.\nu}\det(\nabla^{2}\phi)\ d\underline{t},$$

where $\phi$ is the given Kahler potential. (Notice, by the way, that Holder’s inequality shows that $\nu\mapsto-\log a_{\nu}$ is a convex function, in the obvious sense.) Rescaling, we get $a_{\nu,k}^{-1}=I_{\nu}(k)$ say, where

$$I_{\nu}(k)=\int_{{\bf R}^{n}}e^{-k(\phi-\underline{t}.\nu)}\det(\nabla^{2}\phi)d\underline{t}.$$

(8)

(Notice that these formulae make sense for any $\nu\in\overline{P}$ and the restriction to the lattice $k^{-1}{\bf Z}^{n}$ is not really relevant here.) So we see that our question reduces to the standard discussion of the asymptotic behaviour of the integral * as $k\rightarrow\infty$. The dominant contribution comes from the a neighbourhood of the point $\underline{t}_{0}$ where $\phi-\underline{t}.\nu$ is minimal and the standard Laplace approximation is

$$I_{\nu}(k)\sim(2\pi k)^{-n/2}{\rm exp}(-k(\phi(t_{0})-t_{0}\nu))\det\nabla^{2}\phi(\underline{t}_{0}).$$

But $\underline{t}_{0}$ is just the point which corresponds to $\nu$ under the Legendre transform, and $\phi(\underline{t}_{0})-\underline{t}_{0}.\nu$ is $-u(\nu)$. So

$$k^{-1}\log I_{\nu}(k)=u(\nu)+O(k^{-1}\log k),$$

and our result follows since $k^{-1}\log k\rightarrow 0$ as $k\rightarrow\infty$.

Following on this line, it is easy to derive a special case of Tian’s Theorem from [29]. If we start with any Kahler metric with potential $\phi$, then use the $a_{\nu}^{(k)}$ as above to define an algebraic metric with potential $\phi^{(k)}$ then, after suitable normalisation the $\phi^{(k)}$ converge to $\phi$ as $k\rightarrow\infty$. In particular the algebraic metrics are dense in the space of all metrics.