TITLE: Symplectic form/Kahler metric on a toric manifold
QUESTION [7 upvotes]: I have a standard question about symplectic forms on toric manifolds:
Let $P$ be an $n$-dimensional Delzant polytope and let $X_P$ be the corresponding symplectic toric manifold with symplectic form $\omega_P$ (usual Delzant construction via symplectic reduction). 
In the algebraic geometry side, we can also construct $X_P$ by embedding the torus $(\mathbb{C}^*)^n$ into a projective space $\mathbb{C}\mathbb{P}^{N_k-1}$ using monomials corresponding to the lattice points in $kP$ for a sufficiently large integer $k$ (here $N_k$ is the number of lattice points in $kP$). Then $X_P$ is the closure of the image of the torus in the projective space $\mathbb{C}\mathbb{P}^{N_k-1}$.
My question is, how one can realize the (canonical) symplectic form $\omega_P$ on $X_P$ as the pull-back of a Fubini-Study Kahler form to $X_P$, for an embedding of $X_P$ in a projective space corresponding to the lattice points in some $k\Delta$?

REPLY [5 votes]: Take $P$ be a polyhedral or polytope as follows. Let $G$ be a Torus with Lie algebra $\mathfrak g$ and integral lattice $\mathbb Z_G$. Take primitive vectors $u_1,u_2,...,u_N\in \mathbb Z_G$ spans $\mathfrak g$ and Let $$P=\{\eta\in\mathfrak g^*\mid<\eta,u_i>-\lambda_i\geq 0 , 1\leq   j\leq N\}$$ . Let $P$ be compact then $M_P=(\mathbb CP^n,k\omega_{FS})//_vK$ the Kahler $G$-space with moment map $\phi_P:M_P\to \mathfrak g^*$ where $k$ is a bound of standard simplex. i.e 
$$\Delta_R=\{l\in (\mathbb R^N)^*\mid <l,e_i>\geq 0 \; \; \text{and}\;\; \sum<l,e_i>\leq k\}$$
where $\{e_1,e_2,...,e_n\}$ is the standard basis of $\mathbb R^n$.
Note that this symplex is image of $\mathbb CP^n$ under the moment map for the standard action of $\mathbb T^n$ with Kahler form of $\mathbb CP^n$  being the appropriate multiple of the standard Fubini-Study form