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Locally conformally Kähler metrics obtained from pseudoconvex shells

Authors: Liviu Ornea and Misha Verbitsky
Journal: Proc. Amer. Math. Soc. 144 (2016), 325-335
MSC (2010): Primary 53C55, 53C25
Published electronically: September 9, 2015
MathSciNet review: 3415599
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Abstract: A locally conformally Kähler (LCK) manifold is a complex manifold $ M$ admitting a Kähler covering $ \tilde {M}$, such that its monodromy acts on this covering by homotheties. A compact LCK manifold is called LCK with potential if its covering admits an automorphic Kähler potential. It is known that in this case $ \tilde {M}$ is an algebraic cone, that is, the set of all non-zero vectors in the total space of an anti-ample line bundle over a projective orbifold. We start with an algebraic cone $ C$, and show that the set of Kähler metrics with potential which could arise from an LCK structure is in bijective correspondence with the set of pseudoconvex shells, that is, pseudoconvex hypersurfaces in $ C$ meeting each orbit of the associated $ \mathbb{R}^{>0}$-action exactly once and transversally. This is used to produce explicit LCK and Vaisman metrics on Hopf manifolds, generalizing earlier work by Gauduchon-Ornea, Belgun and Kamishima-Ornea.

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Liviu Ornea
Affiliation: University of Bucharest, Faculty of Mathematics, 14 Academiei Street, 70109 Bucharest, Romania – and – Institute of Mathematics “Simion Stoilow” of the Romanian Academy, 21, Calea Grivitei Street 010702-Bucharest, Romania

Misha Verbitsky
Affiliation: Laboratory of Algebraic Geometry, National Research University HSE, 7 Vavilova Street, Moscow, Russia, 117312

Keywords: Locally conformally K\"ahler manifold, K\"ahler potential, pseudoconvex, Sasakian manifold, Vaisman manifold, Hopf manifold
Received by editor(s): April 21, 2013
Received by editor(s) in revised form: August 24, 2014
Published electronically: September 9, 2015
Additional Notes: The first author was partially supported by CNCS UEFISCDI, project number PN-II-ID-PCE-2011-3-0118.
The second author was partially supported by RSCF grant 14-21-00053 within AG Laboratory NRU-HSE.
Communicated by: Michael Wolf
Article copyright: © Copyright 2015 American Mathematical Society