Holomorphic submersions of locally conformally K\"ahler manifolds

TL;DR

This paper investigates the structure of locally conformally K"ahler (LCK) manifolds, showing that certain holomorphic submersions imply the manifold is either globally conformally K"ahler or a Vaisman manifold, with implications for product manifolds.

Contribution

It establishes a classification result for LCK manifolds admitting holomorphic submersions with K"ahler fibers, linking them to Vaisman manifolds and ruling out certain product structures.

Findings

LCK manifolds with specific holomorphic submersions are either globally conformally K"ahler or Vaisman.

A product of a non-K"ahler LCK and a K"ahler manifold cannot admit an LCK metric.

Abstract

A locally conformally K\"ahler (LCK) manifold is a complex manifold covered by a K\"ahler manifold, with the covering group acting by homotheties. We show that if such a compact manifold X admits a holomorphic submersion with positive dimensional fibers at least one of which is of K\"ahler type, then X is globally conformally K\"ahler or biholomorphic, up to finite covers, to a Vaisman manifold (i.e. a mapping torus over a circle, with Sasakian fibre). As a consequence, we show that the product between a compact non-K\"ahler LCK and a compact K\"ahler manifold cannot carry a LCK metric.