Cohomology of D-complex manifolds

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.Comment: 22 page

Construction of nice nilpotent Lie groups

“…In fact,W can be obtained by intersectingV ∆ with a finite union of parallel affine spaces in V ∆ . Recalling that rows of M ∆ are parametrized by I ∆ , we have: 2 parametrizes a maximal set of Z 2linearly independent rows of M ∆,2 and J ∆ parametrizes a maximal set of R-linearly independent rows of M ∆ . Set…”

Positive projectively flat manifolds are locally conformally flat-Kähler Hopf manifolds

“…Hoping that the following notation is suggestive rather than confusing, we will also denote the curvature tensor of h as Θ :=: Θ [h] , the first Ricci tensor as Θ (1) [h] :=: tr h Θ [h] , and the second Ricci tensor as Θ…”

“…Our next goal is to define a partition on the class of projectively flat metrics. We first need as lemma the Gauduchon conformal method (which we state without proof, that can be found in [1,2,13,24]), for which it is useful to recall the following notion of Chern scalar curvatures Definition 2.1. A further contraction of the Ricci tensors (2) leads to two distinct types of Chern scalar curvatures, as follows…”

“…Whence, we consider the conformal class of a projectively flat metric {h} and we look at the sign of its Gauduchon degree. By the Gauduchon conformal method [1,2,13,24], there exists in {h} a representative whose scalar curvature with respect to the Chern connection has definite sign which is the same as the sign of the Gauduchon degree. This leads to the natural definition of positive (respectively zero, or negative) projectively flat metric (Definition 2.4).…”

Symplectic manifolds and cohomological decomposition

“…Then we can prove the following result, which relates the subgroups H (r,s) ω (X; R) with their invariant part H (r,s) ω (g; R) (compare it with [1,Proposition 2.4] for almost-D-complex structures in the sense of F. R. Harvey and H. B. Lawson, and also with [17,Theorem 3.4] for almost-complex structures). Proposition 3.3.…”