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.

Suppose, in the general situation, that the sections of $L$ generate the sections of $L^{k}$ so that we have a surjective linear map

$$s^{k}(H^{0}(L))\rightarrow H^{0}(L^{k}).$$

A metric on $H^{0}(L)$ defines a metric on the symmetric power $s^{k}(H^{0}(L))$ in a standard way. Then we can define a metric on $H^{0}(L^{k})$ by identifying it with the orthogonal complement of the kernel of the map above. Then we can use this to define an algebraic Kahler metric on $X$ by the embedding $\iota_{k}:X\rightarrow{\bf P}(H^{0}(L^{k})^{*})$. Now, up to a scale factor, these Kahler metrics are independent of $k$. One way of seeing this is that the embedding $\iota_{k}$ is the composite of $\iota_{1}$£ and the Veronese embedding

$$j:{\bf P}({\bf C}^{N})\rightarrow{\bf P}(s^{k}{\bf C}^{N}),$$

and, up to scale, $j$ is an isometry of the two Fubini-Study metrics.(This is forced by $U(N)$-invariance.) So the same Kahler metric has a whole series of algebraic representations.

Let us see how this works in the toric case. We start with data $a_{\nu}$ on $\overline{P}\cap{\bf Z}^{n}$. Then we can write

$$k\phi=\log\left(\sum a_{\nu}e^{\nu.\underline{t}}\right)^{k}=2\log\sum B_{\mu}e^{\mu.\underline{t}},$$

where the coefficients $B_{\mu}$ are

$$B_{\mu}=\sum_{\nu_{1}+\dots\nu_{k}=\mu}a_{\nu_{1}}a_{\nu_{2}}\dots a_{\nu_{k}}.$$

So if we regard $(a_{\mu})$ as a measure $A$ supported on the lattice points in $\overline{P}$ then the $(B_{\mu})$ represent the $k$-fold convolution $A*\dots*A$, supported on the lattice points in $k\overline{P}$. Now rescale back to the fixed polytope $P$, so we write $b_{\nu}^{(k)}=B_{k\nu}$, for $\nu\in\overline{P}\cap k^{-1}{\bf Z}^{n}$. These define an admissible Kahler potential with Legendre transform $ku$, where $u$ is the Legendre transform of $\phi$. Then on compact subsets of $P$ we claim that

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

(9)

This is essentially the Central Limit theorem, for the convolutions of the discrete measure $A$. By applying a translation we can reduce to calculating at the point $\nu=0\in P$. Changing the coefficients $a_{\nu}$ to $a_{\nu}e^{z.\nu}$, for any fixed $z\in{\bf R}^{n}$, does not change either side of (9), when $\nu=0$, so we can reduce to the case when $\sum a_{\nu}\nu=0$. That is to say, that $\phi$ attains its minimum at the point $\underline{t}=0$. Now we consider the function

$$f(\underline{\theta})=\sum a_{\nu}e^{i\nu.\underline{\theta}}.$$

This is a finite trigonometric polynomial which can be regarded as a function on our compact torus $T$. Then

$$b_{0}^{(k)}=\int_{T}f^{k}d\underline{\theta},$$

and our assertion follows from the stationary phase approximation, since the maximum value of $|f|$ is $\sum a_{\nu}=u(0)$.

Of course $f$ is just the analytic continuation of $e^{\phi}$, for our Kahler potential $\phi$. This makes one wonder if there may be other contexts when it is useful to consider such analytic continuations.

Example For each $k$, the round metric on $S^{2}$ is described as an algebraic metric with the coefficients $a_{\nu}=\left(\begin{array}[]{c}k\\ \nu\end{array}\right)$.

Notice that the asymptotics approximations we have discussed hold uniformly over compact subsets of the open polytope $P$. The discussion near the boundary of $P$ is more delicate, because one gets different asymptotic models. A prototype is the different approximations—normal or Poisson–for the binomial distribution in different regimes.