2009/10/28 by Miguel Abreu, Abreu, Miguel, Rosa Sena‐Dias +2
Mathematics · #Algebraic Geometry and Number Theory #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.DG #math.SG
paper · pdf · doi:10.48550/arxiv.0910.5466
30 pages, 3 figures. Revised version: minor corrections, added two references. Version 3: minor changes, corrected sign mystake, updated references
arxiv created 2011/04/18 · arxiv updated 2011/04/19
In a recent paper Donaldson explains how to use an older construction of Joyce to obtain four dimensional local models for scalar-flat Kahler metrics with a 2-torus symmetry. Using this idea, he recovers and generalizes the Taub-NUT metric by including it in a new family of complete scalar-flat toric Kahler metrics. In this paper we generalize Donaldson's method and construct complete scalar-flat toric Kahler metrics on any symplectic toric 4-manifold with "strictly unbounded" moment polygon. These include the asymptotic locally Euclidean scalar-flat Kahler metrics previously constructed by Calderbank and Singer, as well as new examples of complete scalar-flat toric Kahler metrics which are asymptotic to Donaldson's generalized Taub-NUT metrics. Our construction is in symplectic action-angle coordinates and determines all these metrics via their symplectic potentials. When the first Chern class is zero we obtain a new description of known Ricci-flat Kahler metrics.