TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 348, Number 8, August 1996 COMPACT SELF-DUAL HERMITIAN SURFACES VESTISLAV APOSTOLOV, JOHANN DAVIDOV, AND OLEG MU ˇ SKAROV Abstract. In this paper, we obtain a classification (up to conformal equiv- alence) of the compact self-dual Hermitian surfaces. As an application, we prove that every compact Hermitian surface of pointwise constant holomor- phic sectional curvature with respect to either the Riemannian or the Hermit- ian connection is K¨ahler. 1. Introduction A special feature of oriented Riemannian four-manifolds is the fact that the 2-vectors can be decomposed into self-dual and anti-self-dual components under the action of the Hodge star operator *. As a concequence, the Weyl conformal tensor W splits into two parts W + and W - defined by W ± = 1 2 (W±*W). The tensors W ± are invariant under conformal changes of the metric, and reversing the orientation of the manifold interchanges their roles. An oriented Riemannian four-manifold M is said to be self-dual (resp. anti-self-dual) if W - = 0 (resp. W + = 0). It is well-known that the self-duality property plays an important role in the twistor theory since it can be interpreted as the integrability condition for the Atiyah-Hitchin-Singer almost complex structure on the twistor space of M [1]. The classification (up to conformal equivalence) of the compact self-dual mani- folds is a very difficult problem which has been solved so far under additional curva- ture or topological assumptions [5, 6, 7, 8, 9, 10, 13, 15, 16, 17, 19, 21, 23, 24, 25, 28]. The main purpose of the present paper is to obtain a classification of the compact self-dual Hermitian surfaces. If M is a Hermitian surface, its complex structure fixes an orientation on M and this destroys the symmetry between W + and W - . For example, the action of the complex structure on the 2-vectors gives rise to a decomposition of W + whereas W - remains unaffected. In the self-dual case this can be used to obtain, via the Chern-Weil theory andthe Miyaoka inequality, useful integral inequalities involving the scalar curvature, the *-scalar curvature and the norms of the Lee form and the traceless Ricci tensor of M . The main result in this paper is the following: Theorem 1. Any compact self-dual Hermitian surface M which is not conformally flat is conformally equivalent either to CP 2 with the Fubini-Study metric or to a compact quotient of the unit ball in C 2 with the Bergman metric. Received by the editors December 13, 1994. 1991 Mathematics Subject Classification. Primary 53C55. Research parially supported by the Bulgarian Ministry of Science and Education, contract MM-423/94. c 1996 American Mathematical Society 3051 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use