On the Lee classes of locally conformally symplectic complex surfaces

Vestislav Apostolov (1) , Georges Dloussky (2)

1 CIRGET - Centre interuniversitaire de recherches en géométrie et topologie [Montréal]
2 I2M - Institut de Mathématiques de Marseille

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, 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 co-homology 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 1 dR (S, 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.

Mots clés: CLASS VII0, KAHLER, CLASS-VII0, RANK

Domaines: Variables complexes [math.CV]

Vestislav Apostolov, Georges Dloussky. On the Lee classes of locally conformally symplectic complex surfaces. Journal of Symplectic Geometry, 2018, 16 (4), pp.931-958. ⟨hal-02066791⟩