Summary: BIHERMITIAN SURFACES WITH ODD FIRST BETTI
Abstract. Compact bihermitian surfaces are considered, that is, com-
pact, oriented, conformal four-manifolds 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 Enriques-Kodaira classification. Moreover, it must
be either a special Hopf or an Inoue surface (in the strongly bihermitian
case), or is obtained by blowing-up a minimal, class VII surface with
curves (in the non-strongly bihermitian case).
A compact, connected, oriented, conformal 4-manifold (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-