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SCALAR-FLAT KAHLER METRICS ON NON-COMPACT SYMPLECTIC TORIC 4-MANIFOLDS
 

Summary: .
SCALAR-FLAT KšAHLER METRICS ON NON-COMPACT
SYMPLECTIC TORIC 4-MANIFOLDS
MIGUEL ABREU AND ROSA SENA-DIAS
Abstract. In a recent paper Donaldson [D1] explains how to use an older
construction of Joyce [J] to obtain four dimensional local models for scalar-
flat Kšahler metrics with a 2-torus symmetry. In [D2], using the same idea,
he recovers and generalizes the Taub-NUT metric by including it in a new
family of complete scalar-flat toric Kšahler metrics on R4. In this paper we
generalize Donaldson's method and construct complete scalar-flat toric Kšahler
metrics on any symplectic toric 4-manifold with "strictly unbounded" moment
polygon. These include the asymptotically locally Euclidean scalar-flat Kšahler
metrics previously constructed by Calderbank and Singer [CS], as well as new
examples of complete scalar-flat toric Kšahler 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 Kšahler metrics.
1. Introduction
The problem of finding constant scalar curvature Kšahler metrics has been a

  

Source: Abreu, Miguel - Departamento de Matemática, Instituto Superior Técnico, Universidade Técnica de Lisboa

 

Collections: Mathematics