The condition that a toric manifold $X$ be Fano, with $L=K_{X}^{-1}$ is easily stated in terms of the polytope $P$. There is a preferred “centre” $\nu_{0}\in P$ such that for each face $\lambda_{r}(p_{0})-c_{r}=1$. This follows because the wedge product of the vector fields generating the action is a meromorphic $n$-form on $X$ with a simple pole along each of the divisors corresponding to the faces. Then the inverse is a section of $K_{X}^{-1}$ and is a multiple of the standard basis element $s_{\nu_{0}}$. This centre is also the centre of mass of $(\partial P,d\sigma)$.

In this Section we discuss a Theorem of Wang and Zhu [34].

We will begin by giving a proof which is somewhat different to that of Wang and Zhu (although it borrows ideas from that paper and from [32]), working largely with the symplectic description. We can assume that the centre $\nu_{0}$ is the origin. Given a symplectic potential $u$ we write

$$h=x^{i}u_{i}-u,$$

and

$$L=\log\det\nabla^{2}u.$$

These are smooth functions on $P$ but both tend to infinity at the boundary. Note that $h$ depends on a choice of origin in ${\bf R}^{n}$. Of course $h$ is just the composite of the Kahler potential $\phi$ with the derivative of $u$, mapping $P$ to ${\bf R}^{n}$. The assumption that the toric manifold $X$ be Fano is equivalent to the fact that, for any admissible $u$, the difference $L-h$ is a smooth function on $\overline{P}$. The condition that $u$ describe a Kahler-Ricci soliton is that

$$L-h=\sum c_{i}x^{i},$$

(11)

for constants $c_{i}$ (which of course specify the relevant holomorphic vector field on the Kahler manifold). Just as in our discussion of extremal metrics, it is natural in this context to consider more generally an equation $L-h=A$ for some prescribed smooth function $A$ on $\overline{P}$. Again, much as for the extremal case, there are elementary constraints that we need to impose on $A$. For any symplectic potential $u$ we consider the integrals

$$\int_{P}x^{i}e^{L-h}d\underline{x},$$

for $i=1,\dots,n$. Transforming the integral to the dual space, it becomes

$$\int_{{\bf R}^{n}}\frac{\partial\phi}{\partial t_{i}}e^{-\phi}d\underline{t}=-\int_{{\bf R}^{[}n}\frac{\partial e^{-\phi}}{\partial t_{i}}=0.$$

So a necessary condition that the equation $L-h=A$ has a solution is that, for each $i$,

$$\int_{P}x^{i}e^{A}d\underline{x}=0.$$

(12)

This fixes the constants $c_{i}$ in (11). To see this, consider the function of $\underline{c}\in{\bf R}^{n}$:

$$F(\underline{c})=\int_{P}e^{\sum c_{i}x^{i}}d\underline{x}$$

This is convex and proper (since the origin lies in $P$) and so has a unique critical point. But the derivative of $F$ with respect to $c_{i}$ is

$$\int_{P}x^{i}e^{\sum c_{i}x^{i}}\ d\underline{x}.$$

So the unique critical point of $F$ gives exactly the constants $c_{i}$ required to satisfy the constraint.

In sum, the theorem of Wang and Zhu follows from

An equivalent statement is

For any smooth function $A$ on $\overline{P}$ there are constants $\gamma_{i}$ and an admissible potential $u$ such that $L-h=A+\sum\gamma_{i}x^{i}$. The $\gamma_{i}$ are unique and $u$ is unique up to the addition of a linear function.

The equivalence of the statements follows from the same argument as above.