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.