Locally conformally K\"ahler manifolds with holomorphic Lee field

TL;DR

This paper investigates compact locally conformally Kähler (lcK) manifolds with holomorphic Lee vector fields, establishing conditions under which they are Vaisman and providing examples of non-Vaisman cases.

Contribution

It proves that under certain conditions, lcK manifolds with holomorphic Lee fields are Vaisman, and presents examples of non-Vaisman lcK manifolds with holomorphic Lee fields.

Findings

Vaisman condition when Lee field has constant norm

Vaisman condition when metric is Gauduchon

Existence of non-Vaisman lcK manifolds with holomorphic Lee fields

Abstract

We prove that a compact lcK manifold with holomorphic Lee vector field is Vaisman provided that either the Lee field has constant norm or the metric is Gauduchon (i.e., the Lee field is divergence-free). We also give examples of compact lcK manifolds with holomorphic Lee vector field which are not Vaisman.