Bihermitian surfaces with odd first Betti number

Abstract

Compact bihermitian surfaces are considered, that is, compact, oriented, conformal four-manifolds admitting two distinct compatible 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 of the Enriques-Kodaira classification. Moreover, it must be either a special Hopf or an Inoue surface (in the strongly bihermitian case), or obtained by blowing-up a minimal class VII surface with curves (in the non-strongly bihermitian case).