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Morse-Novikov cohomology of locally conformally Kähler surfaces

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Abstract

We review the properties of the Morse-Novikov cohomology and compute it for all known compact complex surfaces with locally conformally Kähler metrics. We present explicit computations for the Inoue surfaces \(\mathcal {S}^0\), \(\mathcal {S}^+\), \(\mathcal {S}^-\) and classify the locally conformally Kähler (and the tamed locally conformally symplectic) forms on \(\mathcal {S}^0\). We prove the nonexistence of LCK metrics with potential and more generally, of \(d_\theta \)-exact LCK metrics on Inoue surfaces and Oeljeklaus-Toma manifolds.

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Acknowledgements

I am grateful to Liviu Ornea for his encouragement and valuable ideas and suggestions that improved this paper and to Massimiliano Pontecorvo for his beautiful explanations and insight about complex surfaces. Many thanks to Daniele Angella, Nicolina Istrati and Miron Stanciu for very stimulating discussions. I thank Andrei Pajitnov for drawing my attention to the results in the paper [34] and to the anonymous referee for his/her constructive comments and suggestions. I also thank Paolo Piccinni and the University of Rome “La Sapienza” for hospitality during part of the work at this paper.

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Authors and Affiliations

  1. Institute of Mathematics “Simion Stoilow” of the Romanian Academy, 21, Calea Grivitei Street, 010702, Bucharest, Romania

    Alexandra Otiman

  2. Faculty of Mathematics and Computer Science, University of Bucharest, 14 Academiei Str., Bucharest, Romania

    Alexandra Otiman

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  1. Alexandra Otiman
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Correspondence to Alexandra Otiman.

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Partially supported by an Erasmus + fellowship from University of Bucharest.

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Otiman, A. Morse-Novikov cohomology of locally conformally Kähler surfaces. Math. Z. 289, 605–628 (2018). https://doi.org/10.1007/s00209-017-1968-y

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