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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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  • $\begingroup$ are not they equal? up to scaling? $\endgroup$ Mar 28, 2015 at 3:01
  • $\begingroup$ Dear @MohammadF.Tehrani, $(X_P, \omega_{FS})$ and $(X_P, \omega_P)$ are (equivarianlty) symplectomorphic. But they are not equal if I require that the symplectomorphism preserves the symplectic structure as well. $\endgroup$
    – user43696
    Mar 30, 2015 at 16:31
  • $\begingroup$ Have you checked this for $f:\mathbb{P}^1\to \mathbb{P}^{m}$, $[z,w]\to[z^m,z^{m-1}w,\ldots, w^m]$? here, everything is explicitly checkable. $\endgroup$ Mar 30, 2015 at 18:11
  • $\begingroup$ Even for m=2 case of example above, it seems to me that the equality you want does not hold. The coefficient of $dz\wedge d\bar{z}$ in $f^*w_{FS}$ is equal to $[a(4|z|^4+|w|^2)-4|z|^4|w|^2]/a^2$, with $a=(1+|z|^4+|z|^2|w|^2+|w|^4)$, which is different from the corresponding coefficient of $dz\wedge d\bar{z}$ in $w_{FS}$ of $\mathbb{P}^1$. $\endgroup$ Mar 30, 2015 at 21:39
  • $\begingroup$ @MohammadF.Tehrani I agree that the two are not equal. $\endgroup$
    – user43696
    Apr 2, 2015 at 19:19

1 Answer 1

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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

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