# LcK structures with holomorphic Lee vector field on Vaisman-type manifolds

## TL;DR

This paper classifies locally conformally Kähler structures with holomorphic Lee vector fields on Vaisman-type manifolds, providing new examples and general results about lcK metrics with potential.

## Contribution

It offers a complete description of such structures on Vaisman-type manifolds and extends results to more general compact complex manifolds.

## Key findings

- Examples of lcK structures with non-homothetic Lee vector fields
- Any lcK metric with potential and holomorphic Lee vector field admits an invariant positive potential
- Classification of lcK structures with holomorphic Lee vector fields on Vaisman-type manifolds

## Abstract

We give a complete description of all locally conformally K\"ahler structures with holomorphic Lee vector field on a compact complex manifold of Vaisman type. This provides in particular examples of such structures whose Lee vector field is not homothetic to the Lee vector field of a Vaisman structure. More generally, dropping the condition of being of Vaisman type, we show that on a compact complex manifold, any lcK metric with potential and with holomorphic Lee vector field admits a potential which is positive and invariant along the anti-Lee vector field.