Kähler geometry of toric manifolds in symplectic coordinates

Abstract

. A theorem of Delzant states that any symplectic manifold (M,#) of dimension 2n, equipped with an e#ective Hamiltonian action of the standard n-torus T n = R n /2#Z n , is a smooth projective toric variety completely determined (as a Hamiltonian T n -space) by the image of the moment map # : M # R n , a convex polytope P = #(M) # R n . In this paper we show, using symplectic (action-angle) coordinates on PT n , how all #-compatible toric complex structures on M can be e#ectively parametrized by smooth functions on P . We also discuss some topics suited for application of this symplectic coordinates approach to Kahler toric geometry, namely: explicit construction of extremal Kahler metrics, spectral properties of toric manifolds and combinatorics of polytopes. 1. Introduction Kahler geometry can be thought of as a "compatible" intersection of complex and symplectic geometries. Indeed, the triple (M 2n , J, #), with 2n the real dimension of M , is a Kahler manifol...