ILLINOIS JOURNAL OF MATHEMATICS Volume 39, Number 4, Winter 1995 KAHLER CURVATURE IDENTITIES FOR TWISTOR SPACES JOHANN DAVIDOV, OLEG MUKAROV AND GUEO GRANTCHAROV 1. Introduction In order to generalize some results of Kiihler geometry, A. Gray [10] introduced and studied three classes of almost-Hermitian manifolds whose curvature tensor resembles that of Kiihler manifolds. They are defined by the following curvature identities: (here J is the almost-complex structure). These identities are very useful in the study of the action of the unitary group on the space of curvature tensors (cf. [16]) as well as for characterizing the Kiihler manifolds in various classes of almost-Hermitian manifolds (for example, see [9], [10], [15], [17], [18]). By a result of S. Goldberg [9] (see also [10]) every compact almost-Kiihler manifold of class s is Kiihlerian and it is an open question raised by A. Gray [10, Th. 5.3] whether the same is true under the weaker condition s 2. We answer negatively to this question showing that the twistor space of a compact Einstein and self-dual 4-manifold with negative scalar curvature provides an example of a compact non-Kiihler almost-Kiihler manifold of class 2. Recall that the twistor space of an oriented Riemannian 4-manifold M is the (2-sphere) bundle 2 on M whose fibre at any point p M consists of all complex structures on TpM compatible with the metric and the opposite Received July 19, 1993 1991 Mathematics Subject Classification. Primary 53C15, 53C25. Research partially supported by the Bulgarian Ministry of Education, Sciences and Culture, contract MM-54/91. (C) 1995 by the Board of Trustees of the University of Illinois Manufactured in the United States of America 627