Conformal vector fields on lcK manifolds

TL;DR

This paper proves that conformal vector fields on compact lcK manifolds are either Killing or holomorphic, depending on the geometric properties of the manifold and its Kähler cover.

Contribution

It establishes conditions under which conformal vector fields on lcK manifolds are necessarily Killing or holomorphic, linking conformal symmetry to complex geometry.

Findings

Conformal vector fields on compact lcK manifolds are Killing with respect to the Gauduchon metric.

Such vector fields are holomorphic if the Kähler cover is neither flat nor hyperkähler.

The results connect conformal symmetry with the complex structure of lcK manifolds.

Abstract

We show that any conformal vector field on a compact lcK manifold is Killing with respect to the Gauduchon metric. Furthermore, we prove that any conformal vector field on a compact lcK manifold whose K\"ahler cover is neither flat, nor hyperk\"ahler, is holomorphic.