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Let $P$ be a Delzant polytope in $\mathbb R^n$, given by a set of inequalities $\ell_i(x) > 0$ where $\ell_i(x) = \sum_k \mu_k^i x_k - \lambda_i$.

In his paper Donaldson says that the Kahler manifold $M_P$ associated to $P$ (the symplectic manifold of Delzant together with with the canonical Guillemin complex structure I suppose) is Fano if and only if there exist a preferred center $x_0$ in the interior of $P$ such that $\ell_i (x_0) = 1$ for all $i$.

My questions are:

  • Are we talking about being Fano with respect to the "canonical" complex structure?

  • Is there a similar criterion to say, given a potential of the form $$ g(x) = \frac 1 2 \sum_i \ell_i(x) \log \ell_i(x) + h(x) $$ which induces a compatible complex structure $J$, whether $(M_P, J)$ is Fano?



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Maybe this can answer my own question. I read in Abreu (Kahler geometry of toric manifolds in symplectic coordinates) Proposition A.1 that the canonical complex structure $J_P$ is T-biholomorphic to any other compatible toric complex structure $J$. So it looks like that being Fano involves only the symplectic structure. Does this make sense? thanks – David Petrecca Nov 21 '13 at 9:37
Maybe I misunderstand what you are saying, but what about the toric degeneration of the toric Fano manifold $\Sigma_0=\mathbb{P}^1\times \mathbb{P}^1$ to the toric Hirzebruch surface $\Sigma_2$? Surely both of these complex structures are compatible with the $(1,1)$-class coming from an ample divisor that remains ample under specialization. Yet they are definitely not biholomorphic. – Jason Starr Dec 5 '13 at 12:49

Your answer to your first question is correct. All of the toric complex structures are $T$-biholomorphic. Being Fano only depends on the vectors $\mu^i$, which define the fan $\Sigma$ of the underlying toric variety. (Strictly speaking the intersections of the hyperplanes $\ell_i(x)=0$ forming faces of the polytope also specify the fan.) The Fano condition, $K^{-1}>0$, is that the support function, a piecewise linear function, defined by the $\ell_i(x)$ with $\lambda_i =-1$ on $\Sigma$ is strictly upper convex (or more precisely concave with given conventions). See W. Fulton Toric Varieties or T. Oda Convex bodies in alg. geo. This support function characterizes the divisor $K^{-1}$, and that it's strictly convex is the condition for $K^{-1}>0$, i.e. that there is a Kaehler metric $\omega$ with $[\omega] =2\pi c_1(K^{-1})$.

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