Miguel Abreu, Instituto Superior Técnico
Kähler geometry of toric manifolds in symplectic coordinates

A theorem of Delzant states that any symplectic manifold $(M,\omega)$ of dimension $2n$, equipped with an effective Hamiltonian action of the standard $n$-torus $\mathbb{T}^n = \mathbb{R}^{n}/2pi\mathbb{Z}^n$, is a smooth projective toric variety completely determined (as a Hamiltonian $\mathbb{T}^n$-space) by the image of the moment map $\phi:M\to\mathbb{R}^n$, a convex polytope $P=\phi(M)\subset\mathbb{R}^n$. In this talk I will show, using symplectic (action-angle) coordinates on $P\times \mathbb{T}^n$, how all $om$-compatible toric complex structures on $M$ can be effectively parametrized by smooth functions on $P$. I will also discuss some topics suited for application of this symplectic coordinates approach to Kähler toric geometry, namely: explicit construction of extremal Kähler metrics, spectral properties of toric manifolds and combinatorics of polytopes.