# Principal elliptic bundles and compact homogeneous l.c.K. manifolds

## TL;DR

This paper classifies and constructs locally conformal Kähler structures on principal elliptic bundles over complex flag manifolds, providing explicit examples and solutions using Lie theory.

## Contribution

It offers a systematic classification of homogeneous l.c.K. structures and solutions to the Hermitian-Einstein-Weyl equation on compact homogeneous Hermitian manifolds.

## Key findings

- Explicit classification of homogeneous l.c.K. structures.
- Construction of new examples on elliptic bundles and hyperKähler manifolds.
- Solutions to the Hermitian-Einstein-Weyl equation using Lie theory.

## Abstract

In this paper, we provide a systematic and constructive description of Vaisman structures on certain principal elliptic bundles over complex flag manifolds. From this description we explicitly classify homogeneous l.c.K. structures on compact homogeneous Hermitian manifolds using elements of representation theory of complex simple Lie algebras. Moreover, we also describe using Lie theory all homogeneous solutions of the Hermitian-Einstein-Weyl equation on compact homogeneous Hermitian-Weyl manifolds. As an application, we provide a huge class of explicit (nontrivial) examples of such structures on homogeneous Hermitian manifolds, these examples include elliptic bundles over full flag manifolds, elliptic bundles over Grassmannian manifolds, and 8-dimensional locally conformal hyperK\"{a}hler compact manifolds.