Toric Kähler Metrics: Cohomogeneity One Examples of Constant Scalar Curvature in Action-Angle Coordinates

Abstract

In these notes of the mini-course given by the author at the XI-th Conference on Geometry, Integrability and Quantization, Varna, Bulgaria, 2009 after the introduction of the toric Kähler geometry, we present Calabi's family of ${\rm U}(n)$-invariant extremal Kähler metrics in symplectic action-angle coordinates and show that it actually contains, as particular cases, many interesting cohomogeneity one examples of constant scalar curvature.