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On the structure of compact strong HKT manifolds

Beatrice Brienza, Anna Fino, Gueo Grantcharov, Misha Verbitsky

TL;DR

This paper establishes new rigidity and structural results for compact strong HKT and BHE manifolds via a detailed analysis of the Bismut connection, holonomy, and associated foliations. It proves that compact BHE manifolds with full Bismut holonomy must be Kähler and shows a holonomy reduction for non‑Kähler cases; it provides a rigidity result for invariant strong HKT structures on solvmanifolds and a complete classification for those with parallel Bismut torsion, including a Beauville–Bogomolov–type decomposition. The authors introduce the Ricci foliation on hypercomplex manifolds, analyze it for compact simply connected 8‑manifolds to reveal an $h^*$‑action and a constant soliton potential, and deduce a Hopf‑fibration structure in dimension eight. They further develop the theory of HKT potentials, conical hypercomplex structures, and Swann‑type constructions, linking local potentials to global conical symmetry and elucidating conditions under which parallel Bismut torsion occurs. Overall, the work provides a coherent framework that unifies holonomy, torsion, and foliation structures to classify and understand 8‑dimensional strong HKT manifolds and their geometric implications.

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 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.

On the structure of compact strong HKT manifolds

TL;DR

This paper establishes new rigidity and structural results for compact strong HKT and BHE manifolds via a detailed analysis of the Bismut connection, holonomy, and associated foliations. It proves that compact BHE manifolds with full Bismut holonomy must be Kähler and shows a holonomy reduction for non‑Kähler cases; it provides a rigidity result for invariant strong HKT structures on solvmanifolds and a complete classification for those with parallel Bismut torsion, including a Beauville–Bogomolov–type decomposition. The authors introduce the Ricci foliation on hypercomplex manifolds, analyze it for compact simply connected 8‑manifolds to reveal an ‑action and a constant soliton potential, and deduce a Hopf‑fibration structure in dimension eight. They further develop the theory of HKT potentials, conical hypercomplex structures, and Swann‑type constructions, linking local potentials to global conical symmetry and elucidating conditions under which parallel Bismut torsion occurs. Overall, the work provides a coherent framework that unifies holonomy, torsion, and foliation structures to classify and understand 8‑dimensional strong HKT manifolds and their geometric implications.

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 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 -dimensional orbifold.
Paper Structure (9 sections, 22 theorems, 123 equations)

This paper contains 9 sections, 22 theorems, 123 equations.

Table of Contents

  1. introduction
  2. Some remarks on the Bismut connection of strong HKT manifolds
  3. Strong HKT manifolds with parallel Bismut torsion
  4. Ricci foliation of an Hypercomplex manifold
  5. Ricci foliation of strong HKT manifolds
  6. Leaves of Ricci foliation are compact
  7. Characterization of $\nabla^B H=0$
  8. HKT potential and conical hypercomplex structures for strong HKT manifolds
  9. Curvature components

Key Result

Proposition 2.3

SULe2ALS Any compact BHE manifold $(M,J,g)$ is a steady pluriclosed soliton with a unique normalized potential $f$. The vector fields $V:=\frac{1}{2} (\theta^\sharp- \operatorname{grad} f)$ and $JV$ are holomorphic Killing and Bismut parallel and they vanish if and only if $(J,g)$ is Kähler. Moreove

Theorems & Definitions (56)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • proof
  • Theorem 2.7
  • proof
  • Remark 2.1
  • ...and 46 more