On the structure of compact strong HKT manifolds

TL;DR

This paper investigates the geometry of compact strong HKT and BHE manifolds, proving conditions under which they are Kähler, classifying certain structures, and exploring their foliation properties, especially in 8-dimensional cases.

Contribution

It establishes new rigidity results for strong HKT manifolds, classifies those with parallel Bismut torsion, and introduces the Ricci foliation concept for hypercomplex manifolds.

Findings

Compact BHE manifolds with full holonomy are Kähler.

Classification of strong HKT manifolds with parallel Bismut torsion.

8-dimensional strong HKT manifolds are Hopf fibrations over 4D orbifolds.

Abstract

We study the geometry of compact strong HKT and, more generally, compact BHE manifolds. We prove that any compact BHE manifold with full holonomy must be K\"ahler and we establish a similar result for strong HKT manifolds. Additionally, we demonstrate a rigidity theorem for strong HKT structures on solvmanifolds and we completely classify those with parallel Bismut torsion. Finally, we introduce the Ricci foliation for hypercomplex manifolds and analyze its properties for compact, simply connected, 8-dimensional strong HKT manifolds, proving that they are always Hopf fibrations over a compact $4$-dimensional orbifold.