On small deformations of paracomplex manifolds

Abstract

Abstract.A paracomplex structure on a manifold M is an endomorphism K of the tangent bundle TM such that K 2 D I , whose˙1-eigenspaces have the same dimension and are involutive. By using the theory of differential graded Lie algebras, we describe small deformations of paracomplex structures. We also compute the space of invariant small deformations of 4-dimensional nilmanifolds endowed with a fixed paracomplex structure.Mathematics Subject Classification (2010). 53C15, 32G07.

Cohomology of D-complex manifolds

“…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.…”

“…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.…”

“…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.…”