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Conformal duality and compact complex surfaces. (English) Zbl 0571.32017
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.

32J15 Compact complex surfaces
53C55 Global differential geometry of Hermitian and Kählerian manifolds
14J15 Moduli, classification: analytic theory; relations with modular forms
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