On the Lee classes of locally conformally symplectic complex surfaces

Abstract

We prove that the deRham cohomology classes of Lee forms of locally conformally symplectic structures taming the complex structure of a compact complex surface $S$ with first Betti number equal to $1$ is either a non-empty open subset of $H_{dR}^1(S,\mathbb{R})$, or a single point. In the latter case, we show that $S$ must be biholomorphic to a blow-up of an Inoue-Bombieri surface. Similarly, the deRham cohomology classes of Lee forms of locally conformally Kähler structures of a compact complex surface $S$ with first Betti number equal to $1$ is either a non-empty open subset of $H_{dR}^1(S,\mathbb{R})$, a single point or the empty set. We give a characterization of Enoki surfaces in terms of the existence of a special foliation, and obtain a vanishing result for the Lichnerowicz-Novikov cohomology groups on the class $\mathrm{VII}$ compact complex surfaces with infinite cyclic fundamental group.

18 pages, a corrected statement in section 5.2