Rigidity results for non-Kähler Calabi-Yau geometries on threefolds

TL;DR

The paper develops a canonical symmetry reduction for compact non-Kähler Calabi–Yau geometries on real six-manifolds by exploiting a Bismut-Hermitian-Einstein (BHE) structure. In dimension six, the transversal geometry becomes conformally Kähler and is governed by a single scalar PDE for the underlying Kähler structure, with a transverse Einstein–Maxwell-type equation linking curvature, torsion, and the soliton potential. A key outcome is a Bott–Chern cohomology rigidity: if the soliton potential is constant, then $h^{1,1}_{BC}(M)\ge2$, with equality precisely for Bismut-flat quotients of $\SU(2)\times\mathbb{R}\times\mathbb{C}$ or $\SU(2)\times\SU(2)$; this extends Gauduchon–Ivanov rigidity to threefolds. The results connect to Hull–Strominger systems and generalized Kähler solitons, reduce the problem to a four-dimensional transversal geometry, and provide sharp topological constraints via transversal Betti numbers, offering a concrete rigidity classification and guiding PDE existence questions on leaf spaces.

Abstract

We derive a canonical symmetry reduction associated to a compact non-Kähler Bismut-Hermitian-Einstein manifold. In real dimension $6$, the transverse geometry is conformally Kähler, and we give a complete description in terms of a single scalar PDE for the underlying Kähler structure. In the case when the soliton potential is constant, we show that that the Bott-Chern number $h^{1,1}_{BC} \geq 2$, and that equality holds if and only if the metric is Bismut-flat, and hence a quotient of either $\SU(2) \times \mathbb R \times \mathbb C$ or $\SU(2) \times \SU(2)$.

Theorems & Definitions (51)

• Theorem 1.1
• Corollary 1.2
• Remark 1.3
• Lemma 2.1
• proof
• Lemma 2.2: GRFBook, Proposition 8.10
• Definition 2.3
• Lemma 2.4: SU2 Proposition 4.1
• Proposition 2.5: lee2024stability Proposition 2.16
• Proposition 2.6