Tohoku Mathematical Journal
2024
March
SECOND SERIES VOL. 76, NO. 1

Tohoku Math. J.
76 (2024), 105-125

Title LEE CLASSES ON LCK MANIFOLDS WITH POTENTIAL

Author Liviu Ornea and Misha Verbitsky

(Received December 14, 2021, revised June 29, 2022)
Abstract. An LCK manifold is a complex manifold $(M,I)$ equipped with a Hermitian form $\omega$ and a closed 1-form $\theta$, called the Lee form, such that $d\omega=\theta\wedge\omega$. An LCK manifold with potential is an LCK manifold with a positive Kähler potential on its universal cover, such that the deck group multiplies the Kähler potential by a constant. A Lee class of an LCK manifold is the cohomology class of the Lee form. We determine the set of Lee classes on LCK manifolds admitting an LCK structure with potential, showing that it is an open half-space in $H^1(M,{\mathbb R})$. For Vaisman manifolds, this theorem was proven in 1994 by Tsukada; we give a new self-contained proof of his result.

Mathematics Subject Classification. Primary 53C55; Secondary 32G05.
Key words and phrases. Locally conformally Kähler, LCK potential, Vaisman manifold, deformation, Lee form, Lee class, Hodge decomposition, algebraic cone, Teichmüller space.

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