Rigidity results for non-K\"ahler Calabi-Yau geometries on threefolds

TL;DR

This paper investigates the structure of non-K"ahler Calabi-Yau geometries on threefolds, deriving symmetry reductions and characterizing special cases through scalar PDEs and topological invariants.

Contribution

It provides a canonical symmetry reduction for non-K"ahler Bismut-Hermitian-Einstein manifolds and characterizes Bismut-flat metrics via Bott-Chern numbers.

Findings

Transverse geometry is conformally K"ahler in real dimension 6.

A scalar PDE describes the underlying K"ahler structure.

Bismut-flat metrics occur only when Bott-Chern number h^{1,1}_{BC} equals 2.

Abstract

We derive a canonical symmetry reduction associated to a compact non-K\"ahler Bismut-Hermitian-Einstein manifold. In real dimension $6$, the transverse geometry is conformally K\"ahler, and we give a complete description in terms of a single scalar PDE for the underlying K\"ahler 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 $\SU(2) \times \SU(2)$.