  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.
