Cohomology of D-complex manifolds

TL;DR

This paper investigates the cohomological properties of D-complex manifolds, establishing a decomposition of the second de Rham cohomology in certain cases and analyzing the stability of D-Kähler structures under deformations.

Contribution

It introduces a cohomological decomposition for 4-dimensional D-complex nilmanifolds and examines the stability of D-Kähler structures under small deformations.

Findings

Decomposition of second de Rham cohomology on 4D D-complex nilmanifolds.

D-Kähler structures are not stable under small deformations.

Provides cohomological tools for D-complex geometry.

Abstract

In order to look for a well-behaved counterpart to Dolbeault cohomology in D-complex geometry, we study the de Rham cohomology of an almost D-complex manifold and its subgroups made up of the classes admitting invariant, respectively anti-invariant, representatives with respect to the almost D-complex structure, miming the theory introduced by T.-J. Li and W. Zhang in [T.-J. Li, W. Zhang, Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds, Comm. Anal. Geom. 17 (2009), no. 4, 651-684] for almost complex manifolds. In particular, we prove that, on a 4-dimensional D-complex nilmanifold, such subgroups provide a decomposition at the level of the real second de Rham cohomology group. Moreover, we study deformations of D-complex structures, showing in particular that admitting D-Kaehler structures is not a stable property under small deformations.