Positivity of LCK Potential
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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).
KeywordsLocally conformally Kähler Potential Plurisubharmonic Holomorphicaly convex Regularized maximum
Mathematics Subject Classification53C55 32E05 32E10
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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