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Twisted differentials and Lee classes of locally conformally symplectic complex surfaces

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Abstract

We study the set of deRham classes of Lee 1-forms of the locally conformally 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.

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Authors and Affiliations

  1. Department of Mathematics, Laboratoire Jean Leray, 2, Rue de la Houssinière, BP 92208, 44322, Nantes, France

    Vestislav Apostolov

  2. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Vestislav Apostolov

  3. Aix-Marseille University, CNRS UMR 7373, I2M, 39 rue F. Joliot-Curie 13453, Marseille Cedex 13, France

    Georges Dloussky

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  1. Vestislav Apostolov
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Correspondence to Vestislav Apostolov.

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To the memory of Marco Brunella whose work inspired us.

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V.A. was supported in part by a “Connect Talent” Grant of the Région des Pays de la Loire in France. He is also grateful to the Institute of Mathematics and Informatics of the Bulgarian Academy of Sciences where a part of this project was realized. G.D. thanks the University of Nantes for their support and hospitality. The authors warmly thank S. Dinew and A. Zeriahi for their advise on the theory of PSH functions. V.A. is grateful to B. Chantraine for his interest in our work and valuable discussions on the topology of LCS manifolds.

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Apostolov, V., Dloussky, G. Twisted differentials and Lee classes of locally conformally symplectic complex surfaces. Math. Z. 303, 76 (2023). https://doi.org/10.1007/s00209-023-03242-5

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