ON THE BOUNDARY BEHAVIOUR OF FRIDMAN INVARIANTS

We prove that the Fridman invariant defined using the Carathéodory pseudodistance does not always go to 1 near strongly Levi pseudoconvex boundary points and it always goes to 0 near nonpseudoconvex boundary points. We also discuss whether Fridman invariants can be extended continuously to some boundary points of domains constructed by deleting compact subsets from other domains.

Homogeneous almost complex manifolds and their compact quotients

This paper investigates the (non)existence of compact quotients of the homogeneous almost-complex strongly-pseudoconvex manifolds discovered and classified by Gaussier–Sukhov [Wong–Rosay theorem in almost complex manifolds, http:www.arXiv.org:math.CV/0307335 ; On the geometry of model almost complex manifolds with boundary, Math. Z. 254(3) (2006) 567–589] and Lee [Domains in almost complex manifolds with an automorphism orbit accumulating at a strongly pseudoconvex boundary point, Michigan Math. J. 54(1) (2006) 179–205; Strongly pseudoconvex homogeneous domains in almost complex manifolds, J. Reine Angew. Math. 623 (2008) 123–160].

APPROXIMATION OF HOLOMORPHIC MAPPINGS ON 1-CONVEX DOMAINS

Let π : Z → X be a holomorphic submersion of a complex manifold Z onto a complex manifold X and D ⋐ X a 1-convex domain with strongly pseudoconvex boundary. We prove that under certain conditions there always exists a spray of π-sections over [Formula: see text] which has prescribed core, it fixes the exceptional set E of D, and is dominating on [Formula: see text]. Each section in this spray is of class [Formula: see text] and holomorphic on D. As a consequence we obtain several approximation results for π-sections. In particular, we prove that π-sections which are of class [Formula: see text] and holomorphic on D can be approximated in the [Formula: see text] topology by π-sections that are holomorphic in open neighborhoods of [Formula: see text]. Under additional assumptions on the submersion we also get approximation by global holomorphic π-sections and the Oka principle over 1-convex manifolds. We include an application to the construction of proper holomorphic maps of 1-convex domains into q-convex manifolds.

SUR L'ÉQUATION DE CAUCHY–RIEMANN TANGENTIELLE DANS UNE CALOTTE STRICTEMENT PSEUDOCONVEXE

We search a cohomological and a geometrical characterization of the open subsets of a strictly pseudoconvex boundary in a Stein manifold on which one can solve the tangential Cauchy–Riemann equation in all bidegrees. On cherche une caractérisation cohomologique et géométrique des ouverts du bord d'un domaine strictement pseudoconvexe relativement compact d'une variété de Stein sur lesquels on peut résoudre l'équation de Cauchy–Riemann tangentielle en tout bidegré.