Lee classes on LCK manifolds with potential

TL;DR

The paper determines the set of Lee classes for compact LCK manifolds with potential, showing these classes form an open half-space in $H^1(M,\mathbb{R})$ determined by a linear functional vanishing on $H^{1,0}(M)\oplus\overline{H^{1,0}(M)}$ and positive on the Lee class. It extends Tsukada’s Vaisman result to higher-dimensional LCK manifolds with potential by establishing a Hodge-type decomposition $H^1(M,\mathbb{C}) = H^{1,0}(M) \oplus \overline{H^{1,0}(M)} \oplus \langle \theta\rangle$, and showing the Lee classes correspond exactly to the positivity set $\mu(\xi)>0$. The work also develops an algebraic-cone perspective via Jordan–Chevalley decomposition and proves that the Kähler cover of a rank-1 LCK manifold with potential is an open algebraic cone; it demonstrates that LCK-with-potential structures can be approximated by Vaisman structures in Teichmüller space, enabling a self-contained proof of the Vaisman case. Consequently, the results clarify the moduli of LCK-with-potential structures, connect complex and algebraic geometry through cone closures, and provide practical criteria for realizing Lee classes in geometric structures.

Abstract

An LCK (locally conformally Kahler) manifold is a complex manifold $(M,I)$ equipped with a Hermitian form $ω$ and a closed 1-form $θ$, called the Lee form, such that $dω=θ\wedgeω$. An LCK manifold with potential is an LCK manifold with a positive Kahler potential on its cover, such that the deck group multiplies the Kahler 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.