2000/04/19 by Miguel Abreu, Abreu, Miguel
Mathematics · #58F05 #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics #Primary 53C55 #Secondary 14M25 #Symplectic Geometry (math.SG) #math.AG #math.DG #math.SG #msc:14M25 #msc:53C55 #msc:58F05
paper · pdf · doi:10.48550/arxiv.math/0004122
24 pages, to appear in "Toric Varieties in Algebraic Geometry and Physics", V. Batyrev (ed.), AMS
arxiv created 2000/04/19 · arxiv updated 2009/11/30
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 \Tn = \Rn/2π\Zn, 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 \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.