$p$-Kähler structures on fibrations and reductive Lie groups

TL;DR

The paper addresses the existence and obstruction of $p$-Kähler and $p$-pluriclosed structures on two main classes: quasi-regular fibrations (notably compact complex homogeneous spaces via the Tits fibration) and invariant complex structures on even-dimensional reductive Lie groups. It develops obstruction criteria based on orbifold line bundle Chern classes and root-system data, showing that under certain positivity and pullback conditions, many fibrations cannot support $(m-j)$-Kähler or $(n-2)$-pluriclosed structures. It then constructs non-regular invariant complex structures on $\mathfrak{sl}(2m-1,\mathbb{R})$ that admit compatible balanced metrics but are not pluriclosed, and contrasts this with a regular family on $\mathfrak{sl}(3,\mathbb{R})$ that also admits balanced metrics. Overall, the results illuminate how algebraic and geometric structures constrain the existence of special Hermitian metrics and provide explicit balanced examples in non-Kähler contexts. These findings advance understanding of the interplay between fibration geometry, Lie-group complex structures, and balanced vs. pluriclosed phenomena.

Abstract

We investigate the existence of $p$-Kähler 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.

Theorems & Definitions (39)

• Definition 2.1
• Definition 2.2
• Definition 2.3: AAAles
• Definition 2.4
• Remark 2.5
• Proposition 2.6: AAAles
• Theorem 3.1
• proof
• Theorem 3.2: Titswang54
• Corollary 3.3