LCK rank of locally conformally Kahler manifolds with potential

TL;DR

This paper studies locally conformally Kahler (LCK) manifolds with potential, showing their LCK rank can vary widely and that manifolds with minimal rank are dense, refining previous results and correcting earlier errors.

Contribution

It demonstrates the possible range of LCK ranks for manifolds with potential and establishes the density of those with minimal rank, advancing understanding of LCK geometry.

Findings

LCK rank can be any integer between 1 and b_1(M)

LCK manifolds with proper potential are dense

Corrects previous inaccuracies in related work

Abstract

An LCK manifold with potential is a compact quotient of a Kahler manifold $X$ equipped with a positive Kahler potential $f$, such that the monodromy group acts on $X$ by holomorphic homotheties and multiplies $f$ by a character. The LCK rank is the rank of the image of this character, considered as a function from the monodromy group to real numbers. We prove that an LCK manifold with potential can have any rank between 1 and $b_1(M)$. Moreover, LCK manifolds with proper potential (ones with rank 1) are dense. Two errata to our previous work are given in the last Section.