Kahler geometry of toric manifolds in symplectic coordinates
Abstract
Description
A theorem of Delzant states that any symplectic manifold $(M,\om)$ of dimension $2n$, equipped with an effective Hamiltonian action of the standard $n$-torus $\T^n = \R^{n}/2π\Z^n$, is a smooth projective toric variety completely determined (as a Hamiltonian $\T^n$-space) by the image of the moment map $ϕ:M\to\R^n$, a convex polytope $P=ϕ(M)\subset\R^n$. In this paper we show, using symplectic (action-angle) coordinates on $P\times \T^n$, how all $\om$-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.
24 pages, to appear in "Toric Varieties in Algebraic Geometry and Physics", V. Batyrev (ed.), AMS
24 pages, to appear in "Toric Varieties in Algebraic Geometry and Physics", V. Batyrev (ed.), AMS