Constant Scalar Curvature and Extremal Kähler Metrics on Blow ups

Abstract

Extremal Kahler metrics were introduced by E. Calabi as best representatives of a given Kahler class of a complex compact manifold, these metrics are critical points of the L 2 norm of the scalar curvature function. In this paper, we report some joint works with C. Arezzo and M. Singer concerning the construction of extremal Kahler metrics on blow ups at finitely many points of Kahler manifolds which already carry an extremal Kahler metric. In particular, we give sufficient conditions on the number and locations of the blown up manifold points for the blow up to carry an extremal Kahler metric.