A ddc-type condition beyond the K\"ahler realm

TL;DR

This paper generalizes the ddc-condition for complex manifolds, providing new characterizations, deformation stability, and invariants that distinguish certain almost complex manifolds from those satisfying the condition.

Contribution

It introduces a new ddc-type condition beyond the Kähler realm, with characterizations, deformation stability, and homotopy invariants that restrict the class of complex manifolds satisfying it.

Findings

The ddc-type condition is hereditary under geometric constructions.

It is an open property under small deformations.

Certain homotopy invariants prevent some manifolds from satisfying the condition.

Abstract

This paper introduces a generalization of the ddc-condition for complex manifolds. Like the dd^c-condition, it admits a diverse collection of characterizations, and is hereditary under various geometric constructions. Most notably, it is an open property with respect to small deformations. The condition is satisfied by a wide range of complex manifolds including all compact complex surfaces, and all compact Vaisman manifolds. We show there are computable invariants of a real homotopy type which in many cases prohibit it from containing any complex manifold satisfying such ddc-type conditions in low degrees. This gives rise to numerous examples of almost complex manifolds which cannot be homotopy equivalent to any of these complex manifolds.