Cohomology of Almost-Complex Manifolds

Int. J. Geom. Methods Mod. Phys.

Villacampa

2014

Section: Introductionmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 99%

Balanced and strongly Gauduchon cones on solvmanifolds

Int. J. Geom. Methods Mod. Phys.

Villacampa

2014

“…Let M be a 2n-dimensional manifold and J an almost complex structure on M . Following [2], [18], let H of bidegrees (p, q) and (q, p) with respect to J, one can consider the subgroups…”

Section: Introductionmentioning

confidence: 99%

“…Also similar subgroups H J (p,q) (M ) and definitions of pure or full almost complex structures can be given by using the space of currents instead of the space of differential forms, and the de Rham homology instead of the de Rham cohomology (for more details and related results see [2], [3], [4], [11], [12], [13], [14], [18]). …”

Section: Introductionmentioning

confidence: 99%

Cohomological decomposition of complex nilmanifolds

2015

TMNA

Abstract. We study pureness and fullness of invariant complex structures on nilmanifolds. We prove that in dimension six, apart from the complex torus, there exist only two non-isomorphic complex structures satisfying both properties, which live on the real nilmanifold underlying the Iwasawa manifold. We also show that the product of two almost complex manifolds which are pure and full is not necessarily full.

“…Angella and Tomassini show in [6] that these properties are no longer true in dimension ≥ 6, by means of two explicit families of curves J t . Since such families J t are not C ∞ -pure for t = 0, they wonder if "more fulfilling counterexamples" could be found.…”

mentioning

confidence: 99%

Cohomological decomposition of compact complex manifolds and holomorphic deformations

2016

Proc. Amer. Math. Soc.

Abstract. The main goal of this note is the study of pureness and fullness properties of compact complex manifolds under holomorphic deformations. Firstly, we construct small deformations of pure-and-full complex manifolds along which one of these properties is lost while the other one is preserved. Secondly, we show that the property of being pure-and-full is not closed under holomorphic deformations. In order to do so, we focus on the class of 6-dimensional solvmanifolds endowed with invariant complex structures. In the special case of nilmanifolds, we also give a classification of those invariant complex structures that are both pure and full. In addition, relations of the cohomological decomposition with other metric and complex properties are studied.