# Condition on moment polytope for a toric manifold to be Fano

Suppose $M$ is a symplectic toric manifold. This means there is a compact torus $T$ that has a Hamiltonian action on $M$, with moment map $\mu:M \to \mathfrak t^*$, and $\dim(M)=2\dim(T)$. Can one tell from the moment polytope $\mu(M)$ whether $M$ is Fano?

• what does it mean for a symplectic manifold to be Fano? or do you want tell if it is Fano for some kahler metric compatible with the symplectic structure? Oct 1, 2015 at 9:32
• @dima-sustretov A symplectic toric manifold can alternately be viewed as a symplectic quotient of a torus acting on $\mathbb C^n$, so it comes with a K\"ahler structure.
– Anon
Oct 1, 2015 at 9:50
• If you can access the book McDuff, Salamon: J-holomorphic curves and symplectic topology, read Section 11.3.1 on page 409. I think the (positive) answer is there. Oct 1, 2015 at 11:25
• It's better to add the condition that $M$ is compact, otherwise you will allow examples like $Bl_0(\mathbb{C}^n)$, which is "Fano" in the sense that $c_1(M)>0$. Also "monotone" is a better word than "Fano" in the symplectic category. Oct 1, 2015 at 18:02

The question is whether the anticanonical class, not the given class $[\omega]$, is ample. Translate the polytope to contain $0$ in the interior. Now translate the facets in/out from the origin, until they're at lattice distance $1$ from the origin. (I.e. a facet is $\{\vec v \in \mathfrak t^*\ :\ \langle \vec v, X \rangle = c > 0\}$ for some unique shortest $X \in \ker(\exp: \mathfrak t {\longrightarrow} T)$; move it to $c=1$). In this way you've computed the moment polytope w.r.t. the anticanonical class = the sum of the toric divisors (each with coefficient $c=1$). The question is whether each old facet still defines a facet of the new polytope.
Non-example: the trapezoid giving $\widetilde{\mathbb P^2}$. Then the intersection of the exceptional curve's facet with the resulting unit triangle is just a corner, not a facet. So no basepoints, but the exceptional curve blows down in the anticanonical non-embedding.