Twisted differentials and Lee classes of locally conformally sympletic complex surfaces

Georges Dloussky (1, 2, 3) , Vestislav Apostolov (4, 5)

1 AMU - Aix Marseille Université
2 CNRS UMR 7373
3 I2M - Institut de Mathématiques de Marseille
4 LMJL - Laboratoire de Mathématiques Jean Leray
5 Institute of Mathematics and Informatics, Bulgarian Academy of Sciences

We study the set of deRham classes of Lee 1-forms of locally cpnformally symplectic (LCS) structures taming the complex structure of a compact complex surface in the Kodaira class VII, and show that the existence of non-trivial upper/lower bounds with respect to the degree function correspond respectively to the existence of certain negative/non-negative PSH functions on the universal cover. We use this to prove that the set of Lee deRham classes of taming LCS is connected, as well as to obtain an explicit negative upper bound for this set on the hyperbolic Kato surfaces. This leads to a complete description of the sets of Lee classes on the known examples of class VII complex surfaces, and to a new obstruction to the existence of bi-hermitian structures on the hyperbolic Kato surfaces of the intermediate type. Our results also reveal a link between bounds of the set of Lee classes and non-trivial logarithmic holomorphic 1-forms with values in a flat holomorphic line bundle.

Mots clés: Kodaira classification+non-Käler surfaces+ Locally conformally symplectic structure

Domaines: Mathématiques [math]

Georges Dloussky, Vestislav Apostolov. Twisted differentials and Lee classes of locally conformally sympletic complex surfaces. Mathematische Zeitschrift, 2023, 303:76, ⟨10.1007/s00209-023-03242-5⟩. ⟨hal-04561228⟩