Compact Complex Surfaces and Constant Scalar Curvature K\"ahler Metrics

Abstract

In this article, I prove the following statement: Every compact complex surface with even first Betti number is deformation equivalent to one which admits an extremal K\"ahler metric. In fact, this extremal K\"ahler metric can even be taken to have constant scalar curvature in all but two cases: the deformation equivalence classes of the blow-up of $\p_2$ at one or two points. The explicit construction of compact complex surfaces with constant scalar curvature K\"ahler metrics in different deformation equivalence classes is given. The main tool repeatedly applied here is the gluing theorem of C. Arezzo and F. Pacard which states that the blow-up/resolution of a compact manifold/orbifold of discrete type, which admits cscK metrics, still admits cscK metrics.