Group actions, non-K\"ahler complex manifolds and SKT structures

TL;DR

This paper constructs new complex structures on principal bundles over complex manifolds, producing non-K"ahler examples and SKT metrics, extending classical constructions and recent results in complex geometry.

Contribution

It generalizes classical complex structure constructions on Lie groups and torus bundles, and introduces new non-K"ahler manifolds with SKT metrics via principal bundle methods.

Findings

Constructed integrable complex structures on principal bundles with compact Lie groups.

Produced large classes of non-K"ahler compact complex manifolds.

Established conditions for the existence of SKT metrics on these bundles.

Abstract

We give a construction of integrable complex structures on the total space of a smooth principal bundle over a complex manifold, with an even dimensional compact Lie group as structure group, under certain conditions. This generalizes the constructions of complex structure on compact Lie groups by Samelson and Wang, and on principal torus bundles by Calabi-Eckmann and others. It also yields large classes of new examples of non-K\"ahler compact complex manifolds. Moreover, under suitable restrictions on the base manifold, the structure group, and characteristic classes, the total space of the principal bundle admits SKT metrics. This generalizes recent results of Grantcharov et al. We study the Picard group and the algebraic dimension of the total space in some cases. We also use a slightly generalized version of the construction to obtain (non-K\"ahler) complex structures on tangential frame bundles of complex orbifolds.