Cohomology of D-complex manifolds

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 Li and Zhang (2009) in [20] 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-Kähler structures is not a stable property under small deformations

Ricci-flat and Einstein pseudoriemannian nilmanifolds

This is partly an expository paper, where the authors' work on pseudoriemannian Einstein metrics on nilpotent Lie groups is reviewed. A new criterion is given for the existence of a diagonal Einstein metric on a nice nilpotent Lie group. Classifications of special classes of Ricci-flat metrics on nilpotent Lie groups of dimension $\leq8$ are obtained. Some related open questions are presented.Comment: 30 pages, 1 figure. v2: added a comment on a recent example of an Einstein nilpotent Lie algebra of dimension 7; added a remark and a question concerning the characteristically nilpotent case; replaced the "\sigma-compatible" condition with the more general "\sigma-diagonal"; added 3 reference