Transformations of locally conformally K\"ahler manifolds

TL;DR

This paper investigates the transformation groups of locally conformally K"ahler manifolds, revealing that certain vector fields must be Killing, holomorphic, and conformal under specific conditions, thereby deepening understanding of their geometric structure.

Contribution

It establishes new results on the nature of conformal and affine vector fields on locally conformally K"ahler manifolds, especially in the context of Vaisman and hyperk"ahler structures.

Findings

All conformal vector fields on certain compact Vaisman manifolds are Killing and holomorphic.

Affine vector fields with respect to the minimal Weyl connection are holomorphic and conformal under specific conditions.

The results clarify the interplay between conformal, affine, and holomorphic transformations in these geometric settings.

Abstract

We consider several transformation groups of a locally conformally K\"ahler manifold and discuss their inter-relations. Among other results, we prove that all conformal vector fields on a compact Vaisman manifold which is neither locally conformally hyperk\"ahler nor a diagonal Hopf manifold are Killing, holomorphic and that all affine vector fields with respect to the minimal Weyl connection of a locally conformally K\"ahler manifold which is neither Weyl-reducible nor locally conformally hyperk\"ahler are holomorphic and conformal