Kähler Geometry of Toric Manifolds in Symplectic Coordinates

A theorem of Delzant states that any symplectic manifold (M,ω) of dimension 2n, equipped with an effective Hamiltonian action of the standard n-torus Tn = Rn/2πZn, is a smooth projective toric variety completely determined (as a Hamiltonian Tn-space) by the image of the moment map φ : M → Rn, a convex polytope P = φ(M) ⊂ Rn. In this paper we show, using symplectic (action-angle) coordinates on P×Tn, how all ω-compatible toric complex structures on M can be effectively parametrized by smooth functions on P . We 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.