On deformations of D-manifolds and CR D-manifolds

Rossi, Federico A.

doi:10.1016/j.geomphys.2011.11.007

Cited by 4 publications

(5 citation statements)

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“…An almost D-complex structure is said integrable (and hence is called D-complex, or also para-complex ) if moreover T + X, T + X ⊆ T + X and T − X, T − X ⊆ T − X , (or, equivalently, if the Nijenhius tensor of K, defined by N K (·, ··) := [·, ··] + [K·, K · ·] − K [K·, ··] − K [·, K · ·] , vanishes). We refer, e.g., to [15,1,7,8,2,3,25,26] and the references therein for more results about (almost) D-complex structures and motivations for their study.…”

Section: D-complex Decompositions Of Cohomology and Homologymentioning

confidence: 99%

“…In this section, we study explicit examples of deformations of D-complex structures on nilmanifolds and solvmanifolds. We refer to [22,25] for more results about deformations of D-complex structures.…”

Section: Small Deformations Of D-complex Structuresmentioning

confidence: 99%

“…vanishes). We refer, e.g., to [15,1,7,8,2,3,25,26] and the references therein for more results about (almost) D-complex structures and motivations for their study.…”

Section: D-complex Decompositions Of Cohomology and Homologymentioning

confidence: 99%

“…Lastly, we study explicit examples of deformations of D-complex structures (see [22,25] for a general account). In particular, we provide examples showing that the dimensions of the D-complex subgroups of the cohomology can jump along a curve of D-complex structures.…”

Section: Introductionmentioning

confidence: 99%

“…Li and W. Zhang.With the aim to write a partial counterpart of [10, Theorem 2.3] in the Dcomplex case, we prove the following result (see §1 for the definition of pure-and-full property).Theorem 3.17. Every invariant D-complex structure on a 4-dimensional nilmanifold is C ∞ -pure-and-full at the 2-nd stage and hence also pure-and-full at the 2-nd stage.Furthermore, we prove that the hypotheses on integrability, nilpotency and dimension can not be dropped out.Lastly, we study explicit examples of deformations of D-complex structures (see [22,25] for a general account). In particular, we provide examples showing that the dimensions of the D-complex subgroups of the cohomology can jump along a curve of D-complex structures.…”

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confidence: 99%

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Cohomology of D-complex manifolds

Angella

Rossi

2012

Differential Geometry and its Applications

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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

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Section: D-complex Decompositions Of Cohomology and Homologymentioning

confidence: 99%

Section: Small Deformations Of D-complex Structuresmentioning

confidence: 99%

“…vanishes). We refer, e.g., to [15,1,7,8,2,3,25,26] and the references therein for more results about (almost) D-complex structures and motivations for their study.…”

Section: D-complex Decompositions Of Cohomology and Homologymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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