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Sep 19, 2013
09/13

by
Andrea Altomani; Costantino Medori; Mauro Nacinovich

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We consider canonical fibrations and algebraic geometric structures on homogeneous CR manifolds, in connection with the notion of CR algebra. We give applications to the classifications of left invariant CR structures on semisimple Lie groups and of CR-symmetric structures on complete flag varieties.

Source: http://arxiv.org/abs/0910.4531v1

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Sep 17, 2013
09/13

by
Judith Brinkschulte; C. Denson Hill; Mauro Nacinovich

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In Grauert's paper [G] it is noted that finite dimensionality of cohomology groups sometimes implies vanishing of these cohomomogy groups. Later on Laufer formulated a zero or infinity law for the cohomology groups of domains in Stein manifolds. In this paper we generalize Laufer's Theorem in [L] and its version for small domains of CR manifolds, proved in [Br], by considering Whitney cohomology on locally closed subsets and cohomology with supports for currents. With this approach we obtain a...

Source: http://arxiv.org/abs/0711.0229v1

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Sep 17, 2013
09/13

by
Judith Brinkschulte; C. Denson Hill; Mauro Nacinovich

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For pseudoconvex abstract CR manifolds, the validity of the Poincare Lemma for (0,1) forms implies local embeddability in C^N. The two properties are equivalent for hypersurfaces of real dimension > or = 5. As a corollary we obtain a criterion for the non validity of the Poicare Lemma for (0,1) forms for a large class of abstract CR manifolds of CR codimension larger than one.

Source: http://arxiv.org/abs/0711.0227v1

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Sep 18, 2013
09/13

by
Mauro Nacinovich; Egmont Porten

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Let $M$ be a $CR$ submanifold of a complex manifold $X$. The main result of this article is to show that $CR$-hypoellipticity at $p_0\in{M}$ is necessary and sufficient for holomorphic extension of all germs of $CR$ functions to an ambient neighborhood in $X$. As an application, we obtain that $CR$-hypoellipticity implies the existence of generic embeddings and prove holomorphic extension for a large class of $CR$ manifolds satisfying a higher order Levi pseudoconcavity condition.

Source: http://arxiv.org/abs/1201.1704v1

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2.0

Jun 29, 2018
06/18

by
Mauro Nacinovich; Egmont Porten

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We introduce various notions of q-pseudo-concavity for abstract CR manifolds and we apply these notions to the study of hyoo-ellipticity, maximum modulus principle and Cauchy problems for CR functions.

Topics: Complex Variables, Mathematics

Source: http://arxiv.org/abs/1611.02553

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46

Sep 22, 2013
09/13

by
Andrea Altomani; Mauro Nacinovich

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In this paper we take up the problem of describing the CR vector bundles M over compact standard CR manifolds S, which are themselves standard CR manifolds. They are associated to special graded Abelian extensions of semisimple graded CR algebras.

Source: http://arxiv.org/abs/math/0307184v1

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Jul 20, 2013
07/13

by
Mauro Nacinovich; Egmont Porten

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Let $M$ be a $CR$ submanifold of a complex manifold $X$. The main result of this article is to show that $CR$-hypoellipticity at $p_0\in{M}$ is necessary and sufficient for holomorphic extension of all germs of $CR$ functions to an ambient neighborhood in $X$. As an application, we obtain that $CR$-hypoellipticity implies the existence of generic embeddings and prove holomorphic extension for a large class of $CR$ manifolds satisfying a higher order Levi pseudoconcavity condition.

Source: http://arxiv.org/abs/1107.3374v2

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60

Sep 23, 2013
09/13

by
C. Denson Hill; Mauro Nacinovich

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Let M be a smooth CR manifold of CR dimension n and CR codimension k, which is not compact, but has the local extension property E. We introduce the notion of "elementary pseudoconcavity" for M, which extends to CR manifolds the concept of a "pseudoconcave" complex manifold. This notion is then used to obtain generalizations, to the noncompact case, of the results of our previous paper about algebraic dependence, transcendence degree and related matters for the field K(M) of...

Source: http://arxiv.org/abs/0710.5031v1

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3.0

Jun 29, 2018
06/18

by
Stefano Marini; Mauro Nacinovich

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We define a class of compact homogeneous CR manifolds which are bases of Mostow fibrations having total spaces equal to their canonical complex realizations and Hermitian fibers. This is used to establish isomorphisms between their tangential Cauchy-Riemann cohomology groups and the corresponding Dolbeault cohomology groups of the embeddings.

Topics: Complex Variables, Mathematics

Source: http://arxiv.org/abs/1612.01757

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Sep 23, 2013
09/13

by
C. Denson Hill; Mauro Nacinovich

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Let $M$ be a smooth compact $CR$ manifold of $CR$ dimension $n$ and $CR$ codimension $k$, which has a certain local extension property $E$. In particular, if $M$ is pseudoconcave, it has property $E$. Then the field $\Cal K(M)$ of $CR$ meromorphic functions on $M$ has transcendence degree $d$, with $d\leq n+k$. If $f_1, f_2, \hdots , f_d$ is a maximal set of algebraically independent $CR$ meromorphic functions on $M$, then $\Cal K(M)$ is a simple finite algebraic extension of the field $\Bbb...

Source: http://arxiv.org/abs/0710.5166v1

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60

Sep 17, 2013
09/13

by
C. Denson Hill; Mauro Nacinovich

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We discuss the maximum modulus principle, and weak unique continuation, for CR functions on an abstract almost CR manifold M. We investigate these matters under the assumption of weak pseudoconcavity, and obtain sharp results about propagation along Sussmann leaves.

Source: http://arxiv.org/abs/0711.1548v1

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113

Sep 18, 2013
09/13

by
Andrea Altomani; Costantino Medori; Mauro Nacinovich

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We study, from the point of view of CR geometry, the orbits M of a real form G of a complex semisimple Lie group G in a complex flag manifold G/Q. In particular we characterize those that are of finite type and satisfy some Levi nondegeneracy conditions. These properties are also graphically described by attaching to them some cross-marked diagrams that generalize those for minimal orbits that we introduced in a previous paper. By constructing canonical fibrations over real flag manifolds, with...

Source: http://arxiv.org/abs/math/0611755v2

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Sep 17, 2013
09/13

by
Judith Brinkschulte; C. Denson Hill; Mauro Nacinovich

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We make two tiny corrections to our previous paper with the same title, and also obtain, as a bonus, something new.

Source: http://arxiv.org/abs/0711.0224v1

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39

Sep 23, 2013
09/13

by
C. Denson Hill; Mauro Nacinovich

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We give different proofs and prove new results on the non complete solvability of some systems of complex first order p.d.e.'s, especially related to the analysis on CR manifolds.

Source: http://arxiv.org/abs/1111.1502v2

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Sep 23, 2013
09/13

by
C. Denson Hill; Mauro Nacinovich

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There is an intrinsic notion of what it means for a contact manifold to be the smooth boundary of a Stein manifold. The same concept has another more extrinsic formulation, which is often used as a convenient working hypothesis. We give a simple proof that the two are equivalent. Moreover it is shown that, even though a border always exists, it's germ is not unique; nevertheless the germ of the Dolbeault cohomology of any border is unique. We also point out that any Stein fillable compact...

Source: http://arxiv.org/abs/0710.5174v1

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87

Jul 19, 2013
07/13

by
C. Denson Hill; Mauro Nacinovich

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In this paper we generalize Leray's calculus of residues in several complex variables, to the situation of an abstract smooth CR manifold M of general type (n,k).

Source: http://arxiv.org/abs/1005.4205v1

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Sep 21, 2013
09/13

by
Andrea Altomani; Costantino Medori; Mauro Nacinovich

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We compute the Euler-Poincar\'e characteristic of the homogeneous compact manifolds that can be described as minimal orbits for the action of a real form in a complex flag manifold.

Source: http://arxiv.org/abs/0709.2608v2

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Sep 23, 2013
09/13

by
C. Denson Hill; Mauro Nacinovich

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We obtain very sharp results about the lack of validity of the Poincare lemma for the tangential Cauchy Riemann equations, acting on tangential forms, tangential to a CR manifold M of general CR dimension n, and general CR codimension k. This generalizes the classical nonsolvability example of H. Lewy. We also discuss the CR structure on the characteristic bundle to M, due to certain degeneracies in the Levi form. A number of naturally geometrically occuring examples are given.

Source: http://arxiv.org/abs/0710.3573v1

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154

Sep 17, 2013
09/13

by
Judith Brinkschulte; C. Denson Hill; Mauro Nacinovich

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In this paper, we consider the boundary M of a weakly pseudoconvex domain in a Stein manifold. We point out a striking difference between the local cohomology and the global cohomology of M, and illustrate this with an example. We also discuss the first and second Cousin problems, and the strong Poincare problem for CR meromorphic functions on the weakly pseudoconvex boundary M.

Source: http://arxiv.org/abs/0711.0225v1

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Jul 22, 2013
07/13

by
Andrea Altomani; C. Denson Hill; Mauro Nacinovich; Egmont Porten

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We prove a subelliptic estimate for systems of complex vector fields under some assumptions that generalize the essential pseudoconcavity for $CR$ manifolds and H\"ormander's bracket condition for real vector fields. Applications are given to prove the hypoellipticity of first order systems and second order partial differential operators. Finally we describe a class of compact homogeneous CR manifolds for which the distribution of $(0,1)$ vector fields satisfies a subelliptic estimate. v2:...

Source: http://arxiv.org/abs/0807.4857v2

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59

Sep 19, 2013
09/13

by
Andrea Altomani; Costantino Medori; Mauro Nacinovich

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Let \^G be a complex semisimple Lie group, Q a parabolic subgroup and G a real form of \^G. The flag manifold \^G/Q decomposes into finitely many G-orbits; among them there is exactly one orbit of minimal dimension, which is compact. We study these minimal orbits from the point of view of CR geometry. In particular we characterize those minimal orbits that are of finite type and satisfy various nondegeneracy conditions, compute their fundamental group and describe the space of their global CR...

Source: http://arxiv.org/abs/math/0507272v2

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3.0

Jun 29, 2018
06/18

by
Stefano Marini; Mauro Nacinovich

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We discuss the relationship between some groups of tangential CR cohomology of some compact homogeneous CR manifolds and the corresponding Dolbeault cohomology groups of their canonical complex embeddings.

Topics: Complex Variables, Mathematics

Source: http://arxiv.org/abs/1612.00698

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Sep 21, 2013
09/13

by
Andrea Altomani; Costantino Medori; Mauro Nacinovich

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We consider a class of compact homogeneous CR manifolds, that we call $\mathfrak n$-reductive, which includes the orbits of minimal dimension of a compact Lie group $K_0$ in an algebraic homogeneous variety of its complexification $K$. For these manifolds we define canonical equivariant fibrations onto complex flag manifolds. The simplest example is the Hopf fibration $S^3\to\mathbb{CP}^1$. In general these fibrations are not $CR$ submersions, however they satisfy a weaker condition that we...

Source: http://arxiv.org/abs/1106.2779v1

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Sep 17, 2013
09/13

by
Andrea Altomani; Costantino Medori; Mauro Nacinovich

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We investigate the $CR$ geometry of the orbits $M$ of a real form $G_0$ of a complex simple group $G$ in a complex flag manifold $X=G/Q$. We are mainly concerned with finite type, Levi non-degeneracy conditions, canonical $G_0$-equivariant and Mostow fibrations, and topological properties of the orbits.

Source: http://arxiv.org/abs/0711.4484v3