Mauro Nacinovich

Complete nondegenerate locally standard CR manifolds

Abstract. Let M be a complete nondegenerate locally standard CR manifold. We show that a necessary and sufficient condition for M to be compact is that the Lie algebra of its infinitesimal CR automorphisms is semisimple. In general we realize M as a Mostow fibration over a compact CR manifoldB whose universal covering is a Cartesian product of Hermitian symmetric spaces and compact nondegenerate standard CR manifolds.

Classification of Semisimple Levi-Tanaka Algebras (*).

After showing that the partial complex structure is defined by an inner derivation, we give a complete classification of semisimple Levi-Tanaka algebra.

Cauchy problem for overdetermined systems

SuntoSi caratterizzano le nozioni di direzioni caratteristiche, formalmente caratteristiche, di iperbolicità e di evoluzione in una, direzione per sistemi a coefficienti costanti, connesse allo studio delle soluzioni del problema di Cauchy in un semispazio.

Remarks on weakly pseudoconvex boundaries

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.

On the nonvanishing of abstract Cauchy–Riemann cohomology groups

In this paper we prove infinite dimensionality of some local and global cohomology groups on abstract Cauchy–Riemann manifolds.

Standard CR manifolds of codimension 2

We give the classification of all finite dimensional Levi-Tanaka algebras of CR codimension two and construct the corresponding standard homogeneous CR manifolds.

Conormal suspensions of differential complexes

§1 An exact sequence for Whitney functions. §2 A generalized Mayer–Vietoris sequence. §3 Carriers and wedge decomposition. §4 Localization. §5 Wedge decomposition for CR manifolds. §6 The de Rham complex associated to a regular assignment of closed sets. §7 The long exact sequence associated to a regular assignment of closed sets. §8 Differential equivalence. §9 Local isomorphisms for the ∆–complex. §10 Tangential complexes. §11 Conormal suspensions. §12 On the mixed de Rham–Dolbeault complex. §13 Poincare lemma and holomorphic extension for the tangential Cauchy–Riemann complex. §14 Solvability wave front sets. §15 Degree q cohomology for q–pseudoconcave CR manifolds. §16 Solvability wave front sets and wedge decomposition for q–pseudoconcave CR manifolds.

Non Completely Solvable Systems of Complex First Order PDE's

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.

Holomorphic Extension from Weakly Pseudoconcave CR Manifolds

Let M be a smooth locally embeddable CR manifold, having some CR dimension m and some CR codimension d. We find an improved local geometric condition on M which guarantees, at a point p on M, that germs of CR distributions are smooth functions, and have extensions to germs of holomorphic functions on a full ambient neighborhood of p. Our condition is a form of weak pseudoconcavity, closely related to essential pseudoconcavity as introduced in [HN1]. Applications are made to CR meromorphic functions and mappings. Explicit examples are given which satisfy our new condition,but which are not pseudoconcave in the strong sense. These results demonstrate that for codimension d > 1, there are additional phenomena which are invisible when d = 1.

On the topology of minimal orbits in complex flag manifolds

We compute the Euler-Poincare characteristic of the homo- geneous compact manifolds that can be described as minimal orbits for the action of a real form in a complex flag manifold.