 
Summary: BIHERMITIAN SURFACES WITH ODD FIRST BETTI
NUMBER
VESTISLAV APOSTOLOV
Abstract. Compact bihermitian surfaces are considered, that is, com
pact, oriented, conformal fourmanifolds admitting two distinct compat
ible complex structures. It is shown that if the first Betti number is odd
then, with respect to either complex structure, such a manifold belongs
to Class VII in the EnriquesKodaira classification. Moreover, it must
be either a special Hopf or an Inoue surface (in the strongly bihermitian
case), or is obtained by blowingup a minimal, class VII surface with
curves (in the nonstrongly bihermitian case).
1. Introduction
A compact, connected, oriented, conformal 4manifold (M, c) is called a
bihermitian surface if it admits two distinct complex structures Ji, i = 1, 2,
compatible with the conformal structure c and the orientation of M; here
and henceforth distinct means that J1(x) = ±J2(x) at some point x of M.
The triple (c, J1, J2) will be then called a (conformal) bihermitian structure
on M; (c, J1, J2) is strongly bihermitian structure if J1 = ±J2 is satisfied
everywhere on M.
One of the reasons motivating the study of bihermitian conformal struc
