Abstract
Let M be a complex manifold and L an oriented real line bundle on M equipped with a flat connection. A “locally conformally Kähler” (LCK) form is a closed, positive (1,1)-form taking values in L, and an LCK manifold is one which admits an LCK form. Locally, any LCK form is expressed as an L-valued pluri-Laplacian of a function called LCK potential. We consider a manifold M with an LCK form admitting an LCK potential (globally on M), and prove that M admits a positive LCK potential. Then M admits a holomorphic embedding to a Hopf manifold, as shown in Ornea and Verbitsky (Math Ann 348:25–33, 2010).
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Acknowledgements
We are grateful to Victor Vuletescu for an interesting counterexample which stimulated our work on this problem; to Stefan Nemirovski for stimulating discussions and reference to Bremermann; to Cezar Joiţa for a careful reading and editing of the paper; to Matei Toma for communicating us the simple proof of Theorem 5.1; and to Jason Starr for invaluable answers given in Mathoverflow. We thank the anonymous referee for her or his most useful comments and suggestions. Liviu Ornea is partially supported by a Grant of Ministry of Research and Innovation, CNCS - UEFISCDI, Project Number PN-III-P4-ID-PCE-2016-0065, within PNCDI III. Misha Verbitsky is partially supported by the Russian Academic Excellence Project ’5-100” and CNPq - Process 313608/2017-2.
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Ornea, L., Verbitsky, M. Positivity of LCK Potential. J Geom Anal 29, 1479–1489 (2019). https://doi.org/10.1007/s12220-018-0046-y
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DOI: https://doi.org/10.1007/s12220-018-0046-y