Start with a Delzant polytope $P$. Let ${\cal S}$ be the finite set of pairs of

• •

  a vertex $p$ of $P$;

• •

  an ordering $\lambda_{r(i)}$ of the faces containing $p$.

For any two $\sigma=(p,r(\ ))$ and $\sigma^{\prime}=(p^{\prime},r^{\prime}(\ ))$ in ${\cal S}$ there is a unique element $\gamma_{\sigma,\sigma^{\prime}}$ of $\Gamma$ which maps $p$ to $p^{\prime}$ and matches up the corresponding faces. Obviously we have

$$\gamma_{\sigma,\sigma}=1\ ;\ \gamma_{\sigma,\sigma^{\prime}}=\gamma_{\sigma^{\prime},\sigma}^{-1}\ ;\ \gamma_{\sigma,\sigma^{\prime\prime}}=\gamma_{\sigma,\sigma^{\prime}}\circ\gamma_{\sigma^{\prime}\sigma^{\prime\prime}}.$$

Now suppose we have any space $M^{*}$ on which $\Gamma$ acts and $M^{*}$ is a subset of a larger space $M$.We take the product ${\cal S}\times M$ and define a relation

$$(\sigma,m)\sim(\sigma^{\prime},\gamma_{\sigma,\sigma^{\prime}}(m)),$$

for $m\in M^{*}$. The properties above tell us that this is an equivalence relation, so we can take the quotient ${\cal S}\times M/\sim$. In our case we take $M$ to be ${\bf C}^{n}$ and $M^{*}=({\bf C}^{*})^{n}\subset{\bf C}^{n}$. Then $GL(n,{\bf Z})$ acts on $M^{*}$. This is clear if we identify ${\bf C}^{*}$ with ${\bf C}/{\bf Z}$ and hence $M^{*}$ with ${\bf C}^{n}/{\bf Z}^{n}$. In terms of the original description, with co-ordinates $z_{i}$ on ${\bf C}^{n}$, we make a matrix $(a_{ij})$ act on $({\bf C}^{*})^{n}$by

$$z^{\prime}_{i}=\prod z_{j}^{a_{ij}},$$

which is well-defined since the $a_{ij}$ are integers. There is a natural homomorphism from $\Gamma$ to $GL(n,{\bf Z})$ so $\Gamma$ acts on $M^{*}$ via this. Then it is clear from the construction that the quotient $X_{{\rm cx.}}$ is a complex manifold covered by charts $M_{\sigma}$ labelled by elements of $\Sigma$, each chart being a copy of $M={\bf C}^{n}$. The charts for the $n!$ different elements of $\Sigma$ belonging to the same vertex of $P$ have the same image so it suffices just to take one of them. There is an action of the complex torus $T^{n}_{c}$ with a dense orbit, which is the image of any $\{\sigma\}\times M^{*}$. The construction behaves well with respect to restriction to faces, so for each $m$-dimensional face $\Pi$ of $P$ there is a submanifold $X^{\Pi}\subset X_{{\rm cx.}}$ which is an $m$-dimensional complex submanifold with an action of $T^{m}_{c}$ induced from the action on $X_{{\rm cx.}}$. Indeed the orbits of the $T^{n}_{c}$ action on $X_{{\rm cx.}}$ correspond to these faces. In particular the vertices of $P$ correspond to points of $X_{{\rm cx.}}$; the fixed points under the $T^{n}_{c}$ action.

Example When $P$ is the $n$-simplex, as above, the manifold $X_{{\rm cx.}}$ we construct is ${\bf C}{\bf P}^{n}$.

So far we have not used the full strength of the data we began with. For example, we could simply have omitted some vertices of $P$ and run the same construction. We have also thrown away some of the data, through the homomomorphism from $\Gamma$ to $GL(n,{\bf Z})$. First, the fact that the vertices come from a bounded polytope yields the compactness of the space $X_{{\rm cx.}}$ we have defined. We leave this as an exercise for the reader. For the second point, it is indeed the case that if we vary the constants $c_{r}$ slightly (so that we do not introduce or remove any vertices) we get the same complex manifold $X_{{\rm cx.}}$. The extra structure of the specific polytope corresponds to fixing a distinguished cohomology class in $H^{2}(X_{{\rm cx.}};{\bf R})$. This is easiest to see in the case when the polytope is integral. Then the $\gamma_{\sigma\sigma^{\prime}}$ lie in a smaller group $\Gamma_{{\bf Z}}\subset\Gamma$ which is an extension

$${\bf Z}^{n}\rightarrow\Gamma_{{\bf Z}}\rightarrow GL(n,{\bf Z}).$$

We take the trivial complex line bundle $\underline{{\bf C}}$ over $M={\bf C}^{n}$. Then $\Gamma_{{\bf Z}}$ acts on the restriction of $\underline{{\bf C}}$ to $M^{*}$ and the same construction gives a complex line bundle $L\rightarrow X_{{\rm cx.}}$. Furthermore this is an equivariant line bundle for the $T^{n}_{c}$ action. The distinguished cohomology class is just the first Chern class of $L$. In general, when the vertices are not integral we consider the sheaf $Z^{1}$ of closed $1$-forms over $X_{{\rm cx.}}$. We can use the $\gamma_{\sigma\sigma^{\prime}}$ to define a closed $1$-form on $M_{\sigma}\cap M_{\sigma^{\prime}}$ and this yields a Cech cocycle with values in this sheaf. Then the short exact sequence of sheaves

$$0\rightarrow{\bf R}\rightarrow C^{\infty}(X_{{\rm cx.}})\rightarrow Z^{1}\rightarrow 0$$

gives a boundary map from $H^{1}(X_{{\rm cx.}};Z^{1})$ to $H^{2}(X_{{\rm cx.}},{\bf R})$ which defines the distinguished cohomology class. (In fact this cohomology class is not changed if we translate $P$. A more precise statement is that the Delzant polytope $P$ can be recovered from the complex manifold $X$ with a suitable distinguished $T^{n}_{c}$-equivariant cohomology class.)

Example. Consider a vertex $p$ of a Delzant polytope $P$. There is no loss of generality in supposing that $p$ is the origin and that near the origin $P$ agrees with the standard model $\{x^{i}>0\}$. Then, for $\delta>0$, we define $P_{\delta}$ to be the subset of $P$ defined by the additional inequality $\sum x_{i}>\delta$. For small enough $\delta$ this is again a Delzant polytope and the complex manifold $X_{\delta}$ is the blow-up of $X$ at the fixed point corresponding to $P$. The exceptional divisor $E$ is a copy of projective space, associated to the “new” $n-1$-simplex in the boundary of $P_{\delta}$. The manifold does not vary with $\delta$ but the evaluation of the distinguished cohomology class on the standard generator of $H_{2}(E)\subset H_{2}(X_{\delta})$ is $\delta$.

Now we go back to differential geometry. If we have a Kahler metric on $X_{{\rm cx.}}$, its restriction to the open orbit is described by a Kahler potential; a convex function $\phi$ on ${\bf R}^{n}$, as above. Conversely we can define am “admissible” convex function $\phi$ to be one which defines a Kahler metric over the orbit which extends smoothly to the compact manifold. This is a condition on the asymptotic behaviour of $\phi$ at infinity in ${\bf R}^{n}$. The essence of the condition is that $\phi$ is asymptotic to the piecewise linear function

$$\Phi(\underline{t})=\max_{p}p.\underline{t},$$

where $p$ runs over the vertices of the polytope. Thus if we let $\phi_{\lambda}$ be the rescaling $\phi_{\lambda}(\underline{t})=\lambda^{-1}\phi(\lambda\underline{t})$ for $\lambda\in{\bf R}$ then $\phi_{\lambda}\rightarrow\Phi$ (in $C^{0}$)as $\lambda$ tends to infinity. In the model case when $0$ is a vertex and $P$ agrees locally with $\{x^{i}>0\}$ the local complex co-ordinates are $z_{a}=\log\tau_{a}$ and so $|z_{a}|^{2}=e^{t_{a}}$. The admissible condition is that $\phi$ extends to a smooth function of the complex co-ordinates $z_{a}$.

Example The round metric on the $2$-sphere with area $2\pi$ is given by the Kahler potential

$$\phi(t)=\log(1+e^{t}).$$

In terms of a local complex co-ordinate $z$ this is $\log(1+|z|^{2})$.