LCK metrics on elliptic principal bundles

TL;DR

This paper investigates the existence of locally conformally K"ahler metrics on elliptic principal bundles over K"ahler manifolds, establishing conditions based on the linear independence of Chern classes.

Contribution

It proves that elliptic principal bundles with linearly independent Chern classes do not admit LCK metrics, extending previous results on K"ahler and LCK structures.

Findings

Bundles with vanishing Chern classes are K"ahler.

Bundles with 1-dimensional span of Chern classes admit LCK metrics.

Linearly independent Chern classes prevent LCK metric existence.

Abstract

For elliptic principal bundles $\pi:X\ra B$ over K\"ahler manifolds it was shown by Blanchard that $X$ has a K\"ahler metric if and only both Chern classes (with real coefficients) of $\pi$ vanish. For some elliptic principal bundles, when the span of these Chern classes is 1-dimensional, it was shown by Vaisman that $X$ carry locally conformally K\"ahler (LCK, for short) metrics. We show that in the case when the Chern classes are linearly independent, $X$ carries no LCK metric.