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 http://arxiv.org/abs/0803.0985 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?

Thanks

David