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 $\log(1+|z|^{2})$.

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 $[0,1]$,

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

Here we suppose that the Delzant polytope $P$ is integral. We consider all the multiples $k\overline{P}$ for integers $k\geq 0$ and let $B_{k}$ be the set of lattice points

$$B_{k}=k\overline{P}\cap{\bf Z}^{n}.$$

Let the number of points in $B_{k}$ be $N_{k}+1$. We can put all these sets together by considering the cone over $P$

$$cone(P)=\{(\underline{x},y)\in{\bf R}^{n+1}:y\geq 0,\underline{x}\in y\overline{P}\}.$$

The disjoint union of the sets $B_{k}$ can be identified with the set $B=cone(P)\cap{\bf Z}^{n+1}$. Now $B$ is an abelian semi-group under addition and we have a corresponding ring $R$ over ${\bf C}$ with one generator $s_{b}$ for each point of $b\in B$ and relations $s_{b}s_{b^{\prime}}=s_{b+b^{\prime}}$. This is a graded ring, $R=\bigoplus R_{k}$, where $R_{k}$ has a basis $s_{\nu}$ corresponding to the points $\nu$ of $B_{k}$. Further, there is an obvious action of the torus $T_{c}^{n}$ on $R$.

All of these definitions make sense for any convex set $P$. The crucial fact is that when the $P$ is an integral polytope the ring is finitely generated. Thus there is a corresponding projective variety $X_{{\rm alg}}={\rm Proj}(R)$, and the group action on $R$ defines an action on $X_{{\rm alg}}$. Second, if $P$ is Delzant, then $X_{{\rm alg}}$ is smooth and of course this recovers the same complex manifold $X_{{\rm cx.}}$. The vector spaces $R_{k}$ are the sections

$$R_{k}=H^{0}(X_{{\rm cx.}},L^{k})$$

and it is not hard to see that for any $k\geq 1$ the sections give an embedding $X_{{\rm cx.}}\rightarrow{\bf P}(R_{k}^{*})$. From this algebro-geometric point of view the integer $\lambda_{r}(\nu)-c_{r}$, for lattice points $\nu\in\overline{P}$, is the order of vanishing of the section $s_{\nu}$ along the corresponding divisor in $X_{{\rm cx.}}$.

Example Let $P$ be the square $(0,1)^{2}\subset{\bf R}^{2}$. The corresponding manifold is the product $S^{2}\times S^{2}$. The points in $B_{1}$ are the four vertices $p_{0}=(0,0),p_{1}=(0,1),p_{2}=(1,0),p_{3}=(1,1)$ so $R_{1}$ has a corresponding basis $s_{0}s_{1},s_{2},s_{3}$ say. The equation $p_{0}+p_{3}=p_{1}+p_{2}$ goes over to the relation $s_{0}s_{3}=s_{1}s_{2}$. The embedding of $X_{{\rm cx.}}$ in ${\bf P}^{3}$ has image the quadric hypersurface cut out by the equation $Z_{0}Z_{1}-Z_{2}Z_{3}=0$.

When the polytope $P$ is integral but not Delzant the variety $X_{{\rm alg}}$ we construct is singular. If each vertex lies on exactly $n$ codimension-1 faces then $X_{{\rm alg}}$ is an orbifold. Much of the theory, including the differential-geometric constructions, extends easily to this case.

To sum up we have three ways—complex, symplectic and algebraic— of constructing a compact manifold associated to an integral Delzant polytope. From now on we will just denote this by $X$.

A toric manifold $X$ contains a submanifold $X_{{\bf R}}$ of one half the dimension which is a “real form” in the complex picture and Lagrangian in the symplectic picture. To define this from the first point of view we just observe that the action of $\Gamma$ on $M^{*}=({\bf C}^{*})^{n}$ preserves the subset $M_{{\bf R}}^{*}$ of real points. Then we run the same construction. From the symplectic point of view we let $A$ be the subgroup of the real torus $T^{n}$ given by the elements of order $2$, so $A$ is isomorphic to $({\bf Z}/2)^{n}$. Then we consider the subset $A\times P\subset T^{n}\times P$ and check that the closure of this in $X$ is a smooth $n$-dimensional manifold. From the algebro-geometric point of view we simply observe that all our relations are real, so complex conjugation acts on everything and we get a real form of our complex algebraic variety.

This construction is particularly vivid in the symplectic picture [kn:Guil2]. The composite

$$X_{{\bf R}}\rightarrow X\rightarrow\overline{P},$$

is a $2^{n}$-fold covering map over the interior $P\subset\overline{P}$ so we can construct $X_{{\bf R}}$ by taking $2^{n}$ copies of $\overline{P}$ and gluing the boundary components appropriately. The Riemannian metric on $P$ given by the Hessian $u_{ij}$ of an admissible symplectic potential extends to a smooth Riemannian metric on $X_{{\bf R}}$. In particular we get a conformal structure on $X_{{\bf R}}$ and when $n=2$ a Riemann surface structure on the oriented cover of $X_{{\bf R}}$. (The surface $X_{{\bf R}}$ is only itself orientable in the case when $P$ is a rectangle.) For example, if $P$ is the standard triangle in ${\bf R}^{2}$ then $X_{{\bf R}}$ is a real projective plane in $X={\bf C}{\bf P}^{2}$ and can be constructed by gluing four triangles. The oriented cover is $S^{2}$, constructed by gluing eight triangles. In general we get a class of Riemann surfaces obtained by gluing eight polygons. Given a symplectic potential $u$, the induced conformal structure on $\overline{P}$ is equivalent to the standard disc. So if $P$ has $s$ vertices we get an invariant of $u$ in the moduli space ${\cal M}_{s}$ of configurations of $s$ distinct points on $S^{1}={\bf R}{\bf P}^{1}$ modulo the action of $PSL(2,{\bf R})$. This determines the conformal structure of $X_{{\bf R}}$, and is an interesting global invariant of a toric Kahler surface.