Hodge theoretic results for nearly Kähler manifolds in all dimensions

2025/09/08 by Albanese, Michael, Karigiannis, Spiro, Martín-Merchán, Lucía +1
#53C25 #53C26 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2509.06664

Abstract

We generalize to nearly Kähler manifolds of arbitrary dimensions most of the Hodge-theoretic results for nearly Kähler 6-manifolds that were established by Verbitsky. In particular, for a compact nearly Kähler manifold of any dimension, the (appropriately defined) Hodge numbers are related to the Betti numbers in the same way as on a compact Kähler manifold. In the 6-dimensional case, Verbitsky was able to say slightly more using the induced SU(3) structure. We discuss potential extensions of this to twistor spaces over positive scalar curvature quaternionic-Kähler manifolds, which are a particular class of (4n+2)-dimensional nearly Kähler manifolds equipped with a special SU(n) ⋅ U(1) structure.

Citations

• Calibrated Geometry in Hyperkahler Cones, 3-Sasakian Manifolds, and Twistor Spaces
• Generalized Ricci Flow
• The Srni lectures on non-integrable geometries with torsion

Related

• Almost complex Hodge theory
• On nearly Kaehler geometry
• On the structure of nearly pseudo-Kähler manifolds
• Compatible almost complex structures on the Hard Lefschetz condition
• Homogeneous nearly Kähler manifolds