On the Lee classes of locally conformally symplectic complex surfaces

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

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Received 30 November 2016

Accepted 9 January 2018

Published 11 February 2019