4,828 research outputs found
Inequalities \`a la Fr\"olicher and cohomological decompositions
We study Bott-Chern and Aeppli cohomologies of a vector space endowed with
two anti-commuting endomorphisms whose square is zero. In particular, we prove
an inequality \`a la Fr\"olicher relating the dimensions of the Bott-Chern and
Aeppli cohomologies to the dimensions of the Dolbeault cohomologies. We prove
that the equality in such an inequality \`a la Fr\"olicher characterizes the
validity of the so-called cohomological property of satisfying the
-Lemma. As an application, we study cohomological
properties of compact either complex, or symplectic, or, more in general,
generalized-complex manifolds.Comment: to appear in J. Noncommut. Geo
Adapted complex tubes on the symplectization of pseudo-Hermitian manifolds
Let be a pseudo-Hermitian space of real dimension , that
is \RManBase is a \CR-manifold of dimension and is a
contact form on giving the Levi distribution . Let
be the canonical symplectization of and
be identified with the zero section of . Then is a
manifold of real dimension which admit a canonical foliation by
surfaces parametrized by , where p\inM is arbitrary and
is the flow generated by the Reeb vector field associated to the contact form
.
Let be an (integrable) complex structure defined in a neighbourhood
of in . We say that the pair is an {adapted complex tube}
on if all the parametrizations defined above are
holomorphic on .
In this paper we prove that if is an adapted complex tube on
, then the real function on defined by the
condition , for each , is a canonical equation for which satisfies the homogeneous
Monge-Amp\`ere equation .
We also prove that if and are real analytic then the
symplectization admits an unique maximal adapted complex tube.Comment: 6 page
Contact Calabi-Yau manifolds and Special Legendrian submanifolds
We consider a generalization of Calabi-Yau structures in the context of
-Sasakian manifolds. We study deformations of a special class of
Legendrian submanifolds and classify invariant contact Calabi-Yau structures on
5-dimensional nilmanifolds. Finally we generalize to codimension .Comment: 16 pages, no figures. Final version to appear in "Osaka J. Math.
Oka principle for Levi flat manifolds
The name of Oka principle, or Oka-Grauert principle, is traditionally used to
refer to the holomorphic incarnation of the homotopy principle: on a Stein
space, every problem that can be solved in the continuous category, can be
solved in the holomorphic category as well. In this note, we begin the study of
the same kind of questions on a Levi-flat manifold; more precisely, we try to
obtain a classification of CR-bundles on a semiholomorphic foliation of type
(n, 1). Our investigation should only be considered a preliminary exploration,
as it deals only with some particular cases, either in terms of regularity or
bidegree of the bundle, and partial results
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