Summary: CORRECTION TO THE PAPER
BIHERMITIAN STRUCTURES ON COMPLEX SURFACES,
PROC. LONDON MATH. SOC. 79 (1999), 414428
VESTISLAV APOSTOLOV, PAUL GAUDUCHON, AND GUEO GRANTCHAROV
Abstract. We here correct two statements, Theorem 1 and Corollary 2, of the
above-mentioned paper and provide a new argument for Corollary 2. We also
update some parts of the paper in the light of recent major new developments
in the theory of bihermitian structures due to N. Hitchin.
1. Theorem 1
Part (ii) of Theorem 1 contains a mistake which has been spotted by N. Hitchin.
The original statement of Theorem 1 must then be replaced by the following one.
Theorem 1. Let (M, c, J1, J2) be a connected, compact bihermitian conformal 4-
manifold with even first Betti number. Then,
(i) either (c, J1, J2) is strongly bihermitian, in which case the complex surfaces
(M, J1) and (M, J2) are either both complex tori or both K3 surfaces, or
(ii) (M, J1) (or equivalently (M, J2)) is either CP2 or a minimal ruled surface
admitting an effective anti-canonical divisor, or a surface obtained from
them by blowing up points along an effective anti-canonical divisor.
Our mistake was the erroneous assertion appearing on page 420 of our paper
that a minimal ruled surface has a K¨ahler metric of negative total scalar curvature