Scalar-flat Kähler metrics on non-compact symplectic toric 4-manifolds

Abstract

In a recent paper, Donaldson (J. Gökova Geom. Topol. 3:1–8, 2009) explains how to use an older construction of Joyce (Duke Math. J. 77:519–552, 1995) to obtain four dimensional local models for scalar-flat Kähler metrics with a 2-torus symmetry. In another paper (Geom. Funct. Anal. 19:83–136, 2009), using the same idea, he recovers and generalizes the Taub-NUT metric by including it in a new family of complete scalar-flat toric Kähler metrics on \({\mathbb{R}^4}\) . In this article, we generalize Donaldson’s method and construct complete scalar-flat toric Kähler metrics on any symplectic toric 4-manifold with “strictly unbounded” moment polygon. These include the asymptotically locally Euclidean scalar-flat Kähler metrics previously constructed by Calderbank and Singer (Invent. Math. 156:405–443, 2004), as well as new examples of complete scalar-flat toric Kähler 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 Kähler metrics.