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 $k\Delta$?

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