 
Summary: BIHERMITIAN STRUCTURES ON COMPLEX SURFACES
V. APOSTOLOV, P. GAUDUCHON and G. GRANTCHAROV
[Received 13 March 1998ÐRevised 14 September 1998]
1. Introduction
We consider connected, oriented (Riemannian) conformal 4manifolds M; c
admitting two independent compatible integrable almostcomplex structures Ji, for
i 1; 2. Here and henceforth, compatible means that J1 and J2 are orthogonal
with respect to any Riemannian metric g in the conformal class c and induce the
chosen orientation of M, and independent means that J1x T 6J2x for some
point x in M. Then, c; J1; J2 will be called a (conformal) bihermitian structure
on M and M; c; J1; J2 a bihermitian surface.
A bihermitian structure will be called strongly bihermitian if J1x T 6J2x
everywhere.
In the sequel, an integrable almostcomplex structure will occasionally be called
a complex structure.
A Riemannian conformal structure c is called antiselfdual, ASD for short, if
the selfdual part W
of the Weyl tensor vanishes identically. It is an easy
consequence of Theorem 4.1 in [5] that c is ASD if and only if the following
holds: for any point x of M, any compatible complex structure of the tangent
