Compact lcK manifolds with parallel vector fields

TL;DR

This paper classifies compact locally conformally Kähler manifolds with non-trivial parallel vector fields, showing they are either Vaisman or globally conformally Kähler, linked to lower-dimensional Kähler manifolds.

Contribution

It provides a complete classification of such manifolds for dimensions greater than two, explicitly describing their structure.

Findings

Manifolds are either Vaisman or globally conformally Kähler.

Explicit description relates manifolds to lower-dimensional Kähler manifolds.

Classification holds for dimensions greater than two.

Abstract

We show that for $n>2$ a compact locally conformally K\"ahler manifold $(M^{2n},g,J)$ carrying a non-trivial parallel vector field is either Vaisman, or globally conformally K\"ahler, determined in an explicit way by some compact K\"ahler manifold of dimension $2n-2$.