On the Lee classes of locally conformally symplectic complex surfaces

TL;DR

This paper classifies the de Rham cohomology classes of Lee forms for locally conformally symplectic and Kähler structures on certain compact complex surfaces with first Betti number one, revealing their possible configurations and characterizations.

Contribution

It establishes a dichotomy for the cohomology classes of Lee forms on these surfaces and characterizes Enoki surfaces via special foliations, also providing vanishing results for certain cohomology groups.

Findings

Cohomology classes form either a non-empty open set or a single point.

Surfaces with a single point class are biholomorphic to blow-ups of Inoue-Bombieri surfaces.

Vanishing of Lichnerowicz-Novikov cohomology groups on class VII surfaces with infinite cyclic fundamental group.

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 $H^1_{dR}(S, \mathbb R)$, 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 K\"ahler structures of a compact complex surface $S$ with first Betti number equal to $1$ is either a non-empty open subset of $H^1_{dR}(S, \mathbb R)$, 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 ${\rm VII}$ compact complex surfaces with infinite cyclic fundamental group.