$p$-K\"ahler structures on fibrations and reductive Lie groups

TL;DR

This paper explores the existence of $p$-K"ahler structures on specific complex manifolds, focusing on fibrations and reductive Lie groups, and provides new examples of balanced manifolds with explicit structures.

Contribution

It introduces new non-regular complex structures on certain Lie algebras and demonstrates their compatibility with balanced metrics, expanding the class of known balanced manifolds.

Findings

Existence of $p$-K"ahler structures on quasi-regular fibrations and homogeneous spaces.

Construction of non-regular complex structures on $rak{sl}(2m-1, r)$.

These structures admit compatible balanced metrics, providing explicit examples.

Abstract

We investigate the existence of $p$-K\"ahler structures on two classes of complex manifolds: on quasi-regular fibrations, with particular emphasis on complex homogeneous spaces, and on reductive Lie groups endowed with invariant complex structures. In the latter setting, we construct non-regular complex structures on the Lie algebras $\mathfrak{sl}(2m-1,\mathbb{R})$ for $m \ge 2$ and show that these structures admit compatible balanced metrics, providing new explicit examples of balanced manifolds.