LCK metrics on elliptic principal bundles

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.