Riv. Mat. Univ. Parma, Vol. 13, No. 2, 2022

Liviu Ornea [a,b] and Misha Verbitsky [c,d]

Compact homogeneous locally conformally Kähler manifolds are Vaisman. A new proof

Pages: 439-448
Received: 30 November 2021
Accepted: 14 April 2022
Mathematics Subject Classification: 53C55, 53C30
Keywords: Locally conformally Kähler, homogeneous, LCK potential, Vaisman manifold, holomorphic action
Authors address:
[a]: University of Bucharest, Faculty of Mathematics and Informatics, Bucharest, Romania
[b]: Institute of Mathematics “Simion Stoilow” of the Romanian Academy, Bucharest, Romania
[c]: Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Brasil
[d]: Laboratory of Algebraic Geometry, Faculty of Mathematics, National Research University Higher School of Economics, Moscow, Russia

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Abstract: An LCK manifold with potential is a complex manifold with a Kähler potential on its cover, such that any deck transformation multiplies the Kähler potential by a constant multiplier. We prove that any homogeneous LCK manifold admits a metric with LCK potential. This is used to give a new proof that any compact homogeneous LCK manifold is Vaisman.

L. Ornea is partially supported by Romanian Ministry of Education and Research, Program PN-III, Project number PN-III-P4-ID-PCE-2020-0025, Contract 30/04.02.2021.
M. Verbitsky is partially supported by the HSE University Basic Research Program, FAPERJ E-26/202.912/2018 and CNPq - Process 313608/2017-2.

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