Here we start with the product $P\times T^{n}$ with standard co-ordinates $x^{a},\theta_{a}$ as before, except of course that now the $\theta_{a}$ are taken to be “angular” co-ordinates with period $4\pi$. This is a noncompact symplectic manifold with the standard symplectic form $\Omega=\sum dx^{a}d\theta_{a}$ and with Hamiltionian $T^{n}$ action whose moment map is the projection to $P$. The essential point is that this can be compactified to a compact symplectic manifold $X_{{\rm symp}}$ and the moment map extends to a map with image the closure $\overline{P}$. This works in a similar fashion to the complex picture. For example, consider the neighbourhood of a vertex of $P$ which as usual we can take to be the origin, with $P$ locally modelled on $\{x^{i}>0\}$. Then $\Omega$ is the pull-back of the standard form on ${\bf C}^{n}$ under the map

$$(x^{a},\theta_{a})\mapsto(|x_{a}|^{1/2}e^{i\theta_{a}}),$$

We adjoin a neighbourhood of $0$ in ${\bf C}^{n}$ to $P\times T^{n}$ using this map and repeat the construction, modified in the obvious way, for all other boundary points of $P$.

Now of course this symplectic construction describes the same object as the complex construction in the previous section. We return to the discussion of the local differential geometry taking now $Q=P$. We can start with an admissible Kahler potential $\phi$ on ${\bf R}^{n}=V^{*}$. Then its Legendre transform is a function on $P$. Around a vertex, as above, this has the form

$$u=\sum x^{i}\log x^{i}+v,$$

where $v$ is a smooth function (on the manifold with corners). We say that a symplectic potential $u$ is admissible if it is the Legendre transform of an admissible Kahler potential $\phi$. Stated explicitly in terms of $u$ this the requirement of “Guillemin boundary conditions”, which are

1. 1.

   $u$ is a continuous function on $\overline{P}$, smooth in the interior.

2. 2.

   The restriction of $u$ to each face is smooth and strictly convex.

3. 3.

   Let $q$ a boundary point which lies on a codimension $r$ face of $P$, so without loss of generality $q=0$ and $P$ is locally defined by equations $x^{1}>0,\dots x^{r}>0$. Then near $q$

   $$u=\sum_{i=1}^{r}x_{i}\log x_{i}+v$$

   where $v$ is smooth.

It is easy to see that such functions exist. For example we can take the Guillemin function

$$u=\sum_{r}(\lambda_{r}-c_{r})\log(\lambda_{r}-c_{r}).$$

Either way, we get a map from the complex manifold $X_{{\rm cx.}}$ to the symplectic manifold $X_{{\rm symp}}$ which matches up the structures involved.

Example The round metric on $S^{2}$, of area $2\pi$, is defined by the symplectic potential, on the interval $[0,1]$,

$$u(x)=\left(x\log x+(1-x)\log(1-x)\right).$$