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.