Miguel Abreu

A (symplectic) toric variety X, of real dimension 2n, is completely determined by its moment polytope Δ ⊂ ℝ n . Recently Guillemin gave an explicit combinatorial way of constructing "toric" Kähler metrics on X, using only data on Δ. In this paper, differential geometric properties of these metrics are investigated using Guillemin's construction. In particular, a nice combinatorial formula for the scalar curvature R is given, and the Euler–Lagrange condition for such "toric" metrics being extremal (in the sense of Calabi) is proven to be R being an affine function on Δ ⊂ ℝ n . A construction, due to Calabi, of a 1-parameter family of extremal Kähler metrics of non-constant scalar curvature on [Formula: see text] is recast very simply and explicitly using Guillemin's approach. Finally, we present a curious combinatorial identity for convex polytopes Δ ⊂ ℝ n that follows from the well-known relation between the total integral of the scalar curvature of a Kähler metric and the wedge product of the first Chern class of the underlying complex manifold with a suitable power of the Kähler class.

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. 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 &amp;quot;compatible&amp;quot; intersection of complex and symplectic geometries. Indeed, the triple (M 2n , J, #), with 2n the real dimension of M , is a Kahler manifol...

A theorem of E. Lerman and S. Tolman, generalizing a result of T. Delzant, states that compact symplectic toric orbifolds are classified by their moment polytopes, together with a positive integer label attached to each of their facets. In this paper we use this result, and the existence of "global" actionangle coordinates, to give an effective parametrization of all compatible toric complex structures on a compact symplectic toric orbifold, by means of smooth functions on the corresponding moment polytope. This is equivalent to parametrizing all toric Kähler metrics and generalizes an analogous result for toric manifolds. A simple explicit description of interesting families of extremal Kähler metrics, arising from recent work of R. Bryant, is given as an application of the approach in this paper. The fact that in dimension four these metrics are self-dual and conformally Einstein is also discussed. This gives rise in particular to a one parameter family of self-dual Einstein metrics connecting the well known Eguchi-Hanson and Taub-NUT metrics.