Hermitian non-K\"{a}hler structures on products of principal $S^{1}$-bundles over complex flag manifolds and applications in Hermitian geometry with torsion

TL;DR

This paper classifies and constructs explicit examples of Hermitian non-Kähler manifolds and related structures on products of principal $S^{1}$-bundles over complex flag manifolds, using Lie algebra representation theory.

Contribution

It introduces a method to explicitly describe and classify normal almost contact structures and applies this to construct new Hermitian non-Kähler manifolds and examples of CYT and astheno-Kähler structures.

Findings

Explicit classification of normal almost contact structures.

Construction of new Hermitian non-Kähler manifolds from principal $S^{1}$-bundles.

Examples of Calabi-Yau structures with torsion and astheno-Kähler structures.

Abstract

In this paper we provide an explicit description of normal almost contact structures obtained from Cartan-Ehresmann connections (gauge fields) on principal $S^{1}$-bundles over complex flag manifolds. The main feature of our approach is to employ elements of representation theory of complex simple Lie algebras in order to describe and classify these structures. We use these normal almost contact structures to explicitly describe a huge class of compact Hermitian non-K\"{a}hler manifolds obtained from products of principal $S^{1}$-bundles over complex flag manifolds. Moreover, we obtain from our description several concrete examples of 1-parametric families of complex structures on products of principal $S^{1}$-bundles over flag manifolds, these concrete examples generalize the Calabi-Eckmann manifolds. Further, as an application of our main results in the setting of KT structures on toric bundles over flag manifolds, we classify a huge class of explicit examples of Calabi-Yau structures with torsion (CYT) on certain Vaisman manifolds (generalized Hopf manifolds). Also as an application of our main results, we provide several new concrete examples of astheno-K\"{a}hler structures on products of compact homogeneous Sasaki manifolds.