Here we work in the neighbourhood of a point in the free orbit $X_{0}$. We can use the group action to define local co-ordinates. So we have complex co-ordinates

$$\tau_{a}=\frac{1}{2}(t_{a}+i\theta_{a})$$

say. The factor $2$ here will simplify the formulae later. Locally the isometry group acts by translations in the $\theta_{a}$ directions. (Later, when we work globally, the $\theta_{a}$ will become “angular” co-ordinates, with period $4\pi$.) Locally, a Kahler metric is given by $i\partial\overline{\partial}\phi$ for a function $\phi$ of the complex variables $\tau_{a}$. If this function only depends on the real parts $t_{a}$ then the metric will obviously be invariant under translations in the $\theta_{a}$ directions and it is not hard to see that any metric of the kind we are considering arises in this way. Now if we write $\phi=\phi(t_{a})$ then the tensor $i\partial\overline{\partial}\phi$ is just

$$\sum_{ab}\frac{\partial\phi}{\partial t_{a}\partial t_{b}}d\tau_{a}d\overline{\tau}_{b},$$

and this defines a positive Hermitian form if and only if the Hessian matrix of $\phi$ is positive definite; or in other words $\phi$ is a convex function of the real variables $\tau_{a}$. Thus the theory of convex functions on Euclidean spaces is embedded, as this translationally invariant case, in the theory of Kahler geometry. We write $\nabla^{2}\phi$ for the Hessian of $\phi$ and also use index notation $\nabla^{2}\phi=(\phi^{ab})$. The placing of the indices is unconventional but will be convenient later. We write $(\phi_{ab})$ for the inverse matrix. Explicitly the symplectic form $\omega$ is

$$\frac{1}{2}\sum\phi^{ab}dt_{a}\wedge d\theta_{b},$$

and the Riemannian metric is

$$\frac{1}{2}\left(\sum\phi^{ab}dt^{a}dt^{b}+\sum\phi^{ab}d\theta^{a}d\theta^{b}\right).$$

We regard the curvature tensor of this metric as an element of $\Lambda^{2}\otimes\Lambda^{2}$. Then the curvature tensor is

$$\sum R^{abcd}d\tau_{a}d\overline{\tau}_{b}\otimes d\tau_{c}d\overline{\tau}_{d},$$

where

$$R^{abcd}=\phi^{abcd}-\phi^{ac\lambda}\phi^{bd\mu}\phi_{\lambda\mu}.$$

(2)

(Here we use the summation convention over the repeated indices. The third and fourth order derivatives of $\phi$ are written as $\phi^{abc},\phi^{abcd}$ in the obvious way.) This formula for the curvature tensor is just the formula (1), expressed in our current notation.)

We now take a different point of view, following Guillemin [15] and Abreu [1], and also the general scheme outlined in the previous section. Thus we consider an open set in ${\bf R}^{n}\times{\bf R}^{n}$ with linear coordinates $x^{a},\theta_{a}$. More invariantly, we should write the ambient space as $V\times V^{*}$ where $V={\bf R}^{n}$, with coordinates $x^{a}$. We assume the open set has the form $Q\times V^{*}$ where $Q\subset V$ is convex. On this open set we consider the standard symplectic form

$$\Omega=\frac{1}{2}\sum dx^{a}d\theta_{a}.$$

This is preserved by the translations in the $\theta$ variables. More precisely we have a Hamiltonian action of the group $G=V^{*}$ on the symplectic manifold $Q\times V^{*}$ and the moment map is just the projection to $Q$, with components the coordinates $x^{a}$. We consider $G$-invariant almost-complex structures on $Q\times V^{*}$, algebraically compatible with $\Omega$. Now at each point such a structure is specified by a subspace of the complexified cotangent bundle which has a unique basis of the form

$$\epsilon_{a}=d\theta_{a}+Z_{ab}dx^{b},$$

where $(Z_{ab})$ is a symmetric complex matrix with positive definite imaginary part. (This is just the standard description of the Siegel upper half-space $Sp(n,{\bf R})/U(n)$.) So our almost-complex structure is represented by a matrix-valued function $(Z_{ab})$ and $G$-invariance specifies that $Z$ is a function of the variables $x^{a}$. Following our general scheme we should now determine when such an almost-complex structure is integrable. By definition this means that the $2$-forms

$$d\epsilon_{a}=\frac{\partial Z_{ab}}{\partial x^{c}}dx^{c}dx^{b}$$

can be expressed as $\sum\alpha_{ab}\wedge\epsilon_{b}$ and this only happens when all the $d\epsilon_{a}$ are zero (since $d\epsilon_{a}$ does not contain any terms involving $d\theta_{i}$). So the integrability condition is

$$\frac{\partial Z_{ab}}{\partial x^{c}}=\frac{\partial Z_{ac}}{\partial x^{b}}.$$

(3)

Now consider the action of the infinite-dimensional symplectomorphism group. In this situation we need to consider the symplectic diffeomorphisms that commute with the $G$-action. More precisely we want to take the Hamiltonian diffeomorphisms generated by functions that Poisson-commute with the generators of the $G$-action; but these are just the functions of the $x^{i}$ variables. The corresponding group ${\cal G}$ of diffeomorphisms can be identified with smooth functions on $Q$, where a function $f$ acts by taking a point $(\underline{x},\underline{\theta})$ to $(\underline{x},\underline{\theta}+Df)$. This gives an action on the space of almost-complex structures which simply takes $Z_{ab}$ to $Z_{ab}+f_{ab}$, where $f_{ab}$ is the Hessian of $f$.

Now consider the action of ${\cal G}$ on the integrable structures. The condition (3) implies, by the elementary “criterion for an exact differential”, that there are complex-valued functions $it_{a}$ such that

$$Z_{ab}=i\frac{\partial t_{a}}{\partial x^{b}}.$$

The fact that $Z_{ab}$ is symmetric implies, by the same criterion, that there is a single complex valued function $F$ such that $t_{a}=\frac{\partial F}{\partial x^{a}}$, in other words

$$Z_{ab}=\frac{\partial^{2}F}{\partial x^{a}\partial x^{b}}.$$

If we let $f$ be minus the real part of $F$ then the action of $f\in{\cal G}$ takes the structure $(Z_{ab})$ to a new structure with zero real part. So ,taking account of this diffeomorphism group, we can reduce to considering $Z=iY$, with $Y$ real and positive definite. Now the functions $t_{a}$ are real and $\epsilon_{a}=d(t_{a}+i\theta_{a})$ so $t_{a}+i\theta_{a}$ are local complex co-ordinates. (Thus we confirm the Newlander-Nirenberg integrability theorem in this special case.). Write $u$ for the imaginary part of the function $F$ above, so

$$Y_{ab}=\frac{\partial^{2}u}{\partial x^{a}\partial x^{b}}=u_{ab}.$$

Some linear algebra shows that the metric defined by the almost complex structure and the fixed form $\Omega$ is

$$\frac{1}{2}\sum u_{ij}dx^{i}dx^{j}+u^{ij}d\theta_{i}d\theta_{j},$$

(4)

where $(u^{ij})$ is the matrix inverse of the Hessian $(u_{ij})$.

The conclusion of this is that we have another description of the local differential geometry, defined by a convex function $u$ of the variables $x^{a}$. The relation between this picture and that in complex co-ordinates discussed above is just the Legendre transform for convex functions. That is, given a convex function $u$ on $Q\subset V$ we define a function $\phi$ on an open set $Q^{*}\subset V^{*}$ by decreeing that

$$\phi(\underline{t})=\sum x^{a}t_{a}-u(\underline{x}),$$

where the point $\underline{x}\in V$ is the unique point where $Du=\underline{t}$. As is well-known, this transform expresses a symmetric relation between $u$ and $\phi$, so $u$ is the Legendre transform of $\phi$. Further, the Hessian $\phi^{ab}=\frac{\partial^{2}\phi}{\partial t_{a}t_{b}}$ is the inverse of the Hessian $u_{ab}$ of $u$ at the corresponding point. It is easy to see using this that the Legendre transform does give a Kahler potential for the same metric expressed in the complex co-ordinates. Conversely if we start with the complex description and a convex function $\phi$ then the Legendre transform gives the symplectic picture. More invariantly, the map $\underline{x}$ is characterised as the moment map for the action of the group of translations.

Thus we have two natural coordinate systems to use when discussing this local differential geometry, and of course we can transform any formulae from one set-up to the other. Working in the symplectic picture we set

$$F_{ijkl}=u_{ia}u_{jb}\frac{\partial^{2}u^{ab}}{\partial x^{k}\partial x^{l}}.$$

Then one finds that the Riemann curvature tensor is

$$F_{ijkl}\eta^{i}\wedge\eta^{k}\otimes\eta{j}\wedge\eta^{l},$$

(5)

where $\eta^{a}=dx^{a}+iu^{ab}d\theta_{b}$. So the four-index tensor $F$ is essentially the same as the curvature tensor. For example the norm if the Riemann curvature tensor is the same as the natural norm of $F$ i.e.

$$|F|^{2}=\sum F_{ijkl}F_{abcd}u^{ia}u^{jb}u^{kc}u^{ld}.$$

The Ricci tensor is in the same fashion, equivalent to the tensor

$$G_{ij}=F_{ijkl}u^{kl},$$

which can also be expressed as

$$G_{ij}=\frac{\partial^{2}L}{\partial x^{i}\partial x^{j}}$$

where $L=\log\det(u_{ij})$. The scalar curvature is given by another contraction yielding Abreu’s formula

$$S=G_{ij}u^{ij}=\sum_{ij}\frac{\partial^{2}u^{ij}}{\partial x^{i}\partial x^{j}}.$$

(6)

We mentioned in the previous section that in the general case the group of symplectomorphisms does not have a complexification, and this limits the practicality of the symplectic approach to Kahler geometry. But in this special situation there is a complexification of ${\cal G}$: simply the complex valued functions on $Q$ under addition. Further, in it is nearly true that this complexified group ${\cal G}^{c}$ acts on the set of almost complex structures, represented as matrix-valued functions $Z_{ab}$. The “action” is simply to map $Z$ to $Z+\frac{\partial^{2}F}{\partial x^{a}\partial x^{b}}$. It is only a local action because the condition that the imaginary part of $X$ is positive definite could be violated. Our discussion above asserts that all the integrable structures are in a single orbit of this complexified action and the parametrisation by the function $u$ is the parametrisation by an open set in the quotient ${\cal G}^{c}/{\cal G}$. Further, it is easy to verify in this framework that the scalar curvature given by the formula (6) is a moment map for the action of ${\cal G}$ with respect to the natural symplectic structure on the space of almost-complex structures (which is derived from the invariant symplectic form on the Siegel upper half space), see [9].

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.

Suppose we start with some symplectic potential $u$ and corresponding Kahler potential $\phi$. Then $\phi$ can be regarded as a Hermitian metric on the line bundle $L$ over the toric variety. Thus we have a natural $L^{2}$-metric on $H^{0}(X;L)$

$$\|s\|^{2}=\int_{X}|s|^{2}d\mu_{\phi},$$

where the pointwise norm $|s|$ is defined by $\phi$ and $d\mu_{\phi}$ is the volume form of the Kahler metric. Thus, starting with $u$ we get a collection of numbers $a_{\nu}=\|s_{\nu}\|^{-1}$. Now replace $L$ by $L^{k}$, as above. The same symplectic potential $u$ defines a metric on $L^{k}$ and we get a collection of numbers $a_{\nu}^{(k)}$ say, for $\nu\in\overline{P}\cap k^{-1}{\bf Z}^{n}$. One precise statement expressing the general idea above is that for each $\epsilon>0$ and compact subset $K\subset P$ there is a $k_{0}$ such that

$$|u(\nu)-k^{-1}\log a_{\nu}^{(k)}|<\epsilon,$$

once $k\geq k_{0}$, for all $\nu\in K\cap k^{-1}{\bf Z}^{n}$.

The proof of this is very simple. Go back to the case $k=1$ for the moment. Unravelling the definitions, the coefficients $a_{\nu}$ are given by

$$a_{\nu}^{-1}=\int_{{\bf R}^{n}}e^{-\phi}e^{\underline{t}.\nu}\det(\nabla^{2}\phi)\ d\underline{t},$$

where $\phi$ is the given Kahler potential. (Notice, by the way, that Holder’s inequality shows that $\nu\mapsto-\log a_{\nu}$ is a convex function, in the obvious sense.) Rescaling, we get $a_{\nu,k}^{-1}=I_{\nu}(k)$ say, where

$$I_{\nu}(k)=\int_{{\bf R}^{n}}e^{-k(\phi-\underline{t}.\nu)}\det(\nabla^{2}\phi)d\underline{t}.$$

(8)

(Notice that these formulae make sense for any $\nu\in\overline{P}$ and the restriction to the lattice $k^{-1}{\bf Z}^{n}$ is not really relevant here.) So we see that our question reduces to the standard discussion of the asymptotic behaviour of the integral * as $k\rightarrow\infty$. The dominant contribution comes from the a neighbourhood of the point $\underline{t}_{0}$ where $\phi-\underline{t}.\nu$ is minimal and the standard Laplace approximation is

$$I_{\nu}(k)\sim(2\pi k)^{-n/2}{\rm exp}(-k(\phi(t_{0})-t_{0}\nu))\det\nabla^{2}\phi(\underline{t}_{0}).$$

But $\underline{t}_{0}$ is just the point which corresponds to $\nu$ under the Legendre transform, and $\phi(\underline{t}_{0})-\underline{t}_{0}.\nu$ is $-u(\nu)$. So

$$k^{-1}\log I_{\nu}(k)=u(\nu)+O(k^{-1}\log k),$$

and our result follows since $k^{-1}\log k\rightarrow 0$ as $k\rightarrow\infty$.

Following on this line, it is easy to derive a special case of Tian’s Theorem from [29]. If we start with any Kahler metric with potential $\phi$, then use the $a_{\nu}^{(k)}$ as above to define an algebraic metric with potential $\phi^{(k)}$ then, after suitable normalisation the $\phi^{(k)}$ converge to $\phi$ as $k\rightarrow\infty$. In particular the algebraic metrics are dense in the space of all metrics.

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.