On the structure of compact strong HKT manifolds

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 Kähler 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.

💡 Research Summary

The paper investigates the global geometry of compact strong HKT (hyper‑Kähler with torsion) manifolds and, more generally, compact BHE (Bismut‑Hermitian‑Einstein) manifolds. The authors first recall the Bismut connection, its torsion 3‑form H, and the notions of SKT, CYT, and BHE. A BHE manifold is a Hermitian manifold that is both CYT (its Bismut connection has holonomy in SU(n)) and SKT (the torsion 3‑form is closed). For strong HKT manifolds the three complex structures I, J, K share the same Bismut connection, so each Hermitian structure is automatically BHE.

The first major result (Section 2) shows that any compact BHE manifold whose Bismut holonomy is “full” (i.e. equal to SU(n)) must be Kähler. The proof relies on the existence of a unique potential function f and the Lee form θ. The vector field V = ½(θ♯ − grad f) is Bismut‑parallel, holomorphic and Killing. If V is non‑zero, the holonomy reduces to SU(n‑1); therefore a full holonomy forces V = 0, which in turn forces θ and f to be constant, implying dω = 0. The same argument works for compact strong HKT manifolds: because the three Hermitian structures share the same θ and f, the common V reduces the holonomy from Sp(n) to Sp(n‑1). Consequently, a compact strong HKT manifold with full Sp(n) holonomy must be hyper‑Kähler.

In Section 3 the authors classify strong HKT manifolds with parallel Bismut torsion. Parallel torsion implies that the Bismut connection is Ricci‑flat and that the manifold splits, after passing to a finite cover, as a product X × Y where X is a compact hyper‑Kähler manifold and Y is a compact Bismut‑flat manifold. This is a Beauville‑Bogomolov‑type decomposition extending earlier results for BHE manifolds to the hyper‑Hermitian setting.

Section 4 introduces the “Ricci foliation” of a hyper‑complex manifold: the kernel of the Ricci tensor of the Obata connection. The authors study its basic properties and show that it is integrable in many situations.

The most intricate part of the paper (Sections 5–6) deals with compact simply‑connected strong HKT manifolds of real dimension 8. Using the Ricci foliation they prove that the foliation contains an integrable sub‑distribution isomorphic to su(2) ⊕ u(1). This yields an isometric action of H* ≅ S¹ × SU(2) on the manifold. The potential function f is shown to be constant, which forces the Lee form to be parallel. Consequently the manifold admits a foliation with compact leaves, and the leaf space is a compact 4‑dimensional orbifold. The total space is therefore a Hopf fibration over this orbifold. In particular, when the Bismut torsion is parallel, the only possible example (up to finite cover) is the Lie group SU(3) equipped with its left‑invariant strong HKT structure.

Finally, Section 7 discusses the relationship between the HKT potential and the geometry of the Ricci foliation, and the paper ends with remarks on possible extensions (higher dimensions, non‑simply‑connected cases, connections to supersymmetric sigma‑models).

Overall, the work provides a comprehensive rigidity theory for compact strong HKT and BHE manifolds: full holonomy forces Kähler/hyper‑Kähler, parallel torsion forces a product decomposition, and in dimension 8 the Ricci foliation yields a Hopf‑type fibration structure. The techniques combine Bismut connection analysis, potential theory, Lie‑algebraic curvature computations on solvable groups, and foliation theory, offering a clear picture of how torsion, holonomy, and global topology intertwine in these geometries.