Conformal duality and compact complex surfaces (Q1061910)

Let M be a compact complex surface with first Betti number \(b_ 1\) which admits a compatible anti-self-dual metric g. In this paper we prove that if \(b_ 1\) is even, then g is conformally equivalent to a Kähler metric with zero scalar curvature and if \(b_ 1\) is odd then g is conformally equivalent to a locally conformally Kähler metric with positive scalar curvature. Applying this result to the Enriques-Kodaira classification of compact complex surfaces, we give a list of possible surfaces which admit an anti-self-dual metric.