On the existence of compact scalar-flat Kähler surfaces

Abstract

A compact compler surface with non-trivial canonical bundle and a Kähler metric ofzero scalar curvature must be a ruled surface. It is also known that not every ruled surface can admit such extremal Kähler metrics. In this paper we review recentjoint work with Kim and LeBrun in which deforma- tion theory of pairs of singular complex spaces it is used to show that any ruled surface (M, J) has blow-ups (M, J) which admit Kähler metrics of zero scalar curvature.

Keywords

• kähler metric
• twistor space