On the scalar curvature of complex surfaces

On the Scalar Curvature of Complex Surfaces

Let (M, J) be a minimal compact complex surface of Kähler type. It is shown that the smooth 4-manifold M admits a Riemannian metric of positive scalar curvature iff (M,J) admits a Kähler metric of positive scalar curvature. This extends previous results of Witten and Kronheimer. A complex surface is a pair (M,J) consisting of a smooth compact 4-manifold M and a complex structure J on M ; the la...

Constant scalar curvature metrics on toric surfaces

A Remark on Kähler Metrics of Constant Scalar Curvature on Ruled Complex Surfaces

In this note we point out how some recent developments in the theory of constant scalar curvature Kähler metrics can be used to clarify the existence issue for such metrics in the special case of (geometrically) ruled complex surfaces. Dedicated to Professor Paul Gauduchon on his 60th birthday.

Kähler Metrics of Constant Scalar Curvature on Hirzebruch Surfaces

It is shown that a Hirzebruch surface admits a Kähler metric (possibly indefinite) of constant scalar curvature if and only if its degree equals zero. There have been many extensive studies for positive-definite Kähler metrics of constant scalar curvature, especially, Kähler Einstein metrics and scalar-flat Kähler metrics, on existence, uniqueness, obstructions, and relationships with other not...

On the Scalar Curvature of Einstein Manifolds

We show that there are high-dimensional smooth compact manifolds which admit pairs of Einstein metrics for which the scalar curvatures have opposite signs. These are counter-examples to a conjecture considered by Besse [6, p. 19]. The proof hinges on showing that the Barlow surface has small deformations with ample canonical line bundle.