Linearly invariant families of holomorphic mappings of a ball. The dimension reduction method.
Locally biholomorphic mappings of multiconnected domains.
Loewner Chains and Biholomorphic Mappings in Cn and Reflexive Complex Banach Spaces
Manifolds with a unique embedding
We show that if X, Y are smooth, compact k-dimensional submanifolds of ℝⁿ and 2k+2 ≤ n, then each diffeomorphism ϕ: X → Y can be extended to a diffeomorphism Φ: ℝⁿ → ℝⁿ which is tame (to be defined in this paper). Moreover, if X, Y are real analytic manifolds and the mapping ϕ is analytic, then we can choose Φ to be also analytic. We extend this result to some interesting categories of closed (not necessarily compact) subsets of ℝⁿ, namely, to the category of Nash submanifolds...
Méthodes de changement d’échelles en analyse complexe
Nous mettons en perspective différentes méthodes de changement d’échelles et illustrons leur pertinence en mettant sur pieds des preuves simples et élémentaires de plusieurs théorèmes biens connus en analyse ou géométrie complexe. Les situations abordées sont variées et la plupart des théorèmes démontrés sont des classiques initialement obtenus entre la fin du xixe et la seconde moitié du xxe siècle.
Minimizing the Energy of Maps from a surface into a 2-shpere with prescribed degree and boundary values.
Morphisms of projective varieties from the viewpoint of minimal model theory [Book]
Multiplicity and the Łojasiewicz exponent
We give a formula for the multiplicity of a holomorphic mapping , m > n, at an isolated zero, in terms of the degree of an analytic set at a point and the degree of a branched covering. We show that calculations of this multiplicity can be reduced to the case when m = n. We obtain an analogous result for the local Łojasiewicz exponent.
Noguchi-type convergence-extension theorems for (n,d)-sets
We introduce the notion of (n,d)-sets and show several Noguchi-type convergence-extension theorems for (n,d)-sets.
Non-embeddability of general unipotent diffeomorphisms up to formal conjugacy
The formal class of a germ of diffeomorphism is embeddable in a flow if is formally conjugated to the exponential of a germ of vector field. We prove that there are complex analytic unipotent germs of diffeomorphisms at () whose formal class is non-embeddable. The examples are inside a family in which the non-embeddability is of geometrical type. The proof relies on the properties of some linear functional operators that we obtain through the study of polynomial families of diffeomorphisms...
Oka manifolds: From Oka to Stein and back
Oka theory has its roots in the classical Oka-Grauert principle whose main result is Grauert’s classification of principal holomorphic fiber bundles over Stein spaces. Modern Oka theory concerns holomorphic maps from Stein manifolds and Stein spaces to Oka manifolds. It has emerged as a subfield of complex geometry in its own right since the appearance of a seminal paper of M. Gromov in 1989.In this expository paper we discuss Oka manifolds and Oka maps. We describe equivalent characterizations...
On D*-extension property of the Hartogs domains.
A complex analytic space is said to have the D*-extension property if and only if any holomorphic map from the punctured disk to the given space extends to a holomorphic map from the whole disk to the same space. A Hartogs domain H over the base X (a complex space) is a subset of X x C where all the fibers over X are disks centered at the origin, possibly of infinite radius. Denote by φ the function giving the logarithm of the reciprocal of the radius of the fibers, so that, when X is pseudoconvex,...
On holomorphic maps into compact non-Kähler manifolds
We study the extension problem of holomorphic maps of a Hartogs domain with values in a complex manifold . For compact Kähler manifolds as well as various non-Kähler manifolds, the maximal domain of extension for over is contained in a subdomain of . For such manifolds, we define, in this paper, an invariant Hex using the Hausdorff dimensions of the singular sets of ’s and study its properties to deduce informations on the complex structure of .
On invariant domains in certain complex homogeneous spaces
Given a compact connected Lie group . For a relatively compact -invariant domain in a Stein -homogeneous space, we prove that the automorphism group of is compact and if is semisimple, a proper holomorphic self mapping of is biholomorphic.
On locally biholomorphic mappings from multi-connected onto simply connected domains
We continue E. Ligocka's investigations concerning the existence of m-valent locally biholomorphic mappings from multi-connected onto simply connected domains. We decrease the constant m, and also give the minimum of m in the case of mappings from a wide class of domains onto the complex plane ℂ.
On locally biholomorphic surjective mappings
We prove that each open Riemann surface can be locally biholomorphically (locally univalently) mapped onto the whole complex plane. We also study finite-to-one locally biholomorphic mappings onto the unit disc. Finally, we investigate surjective biholomorphic mappings from Cartesian products of domains.
On some rigidity properties of mappings between CR-submanifolds in complex space
We survey some recent results on holomorphic or formal mappings sending real submanifolds in complex space into each other. More specifically, the approximation and convergence properties of formal CR-mappings between real-analytic CR-submanifolds will be discussed, as well as the corresponding questions in the category of real-algebraic CR-submanifolds.
On some specific nonlinear ordinary difference equations
On the automorphism group of strongly pseudoconvex domains in almost complex manifolds
In contrast with the integrable case there exist infinitely many non-integrable homogeneous almost complex manifolds which are strongly pseudoconvex at each boundary point. All such manifolds are equivalent to the Siegel half space endowed with some linear almost complex structure.We prove that there is no relatively compact strongly pseudoconvex representation of these manifolds. Finally we study the upper semi-continuity of the automorphism group of some hyperbolic strongly pseudoconvex almost...
On the automorphisms of the spectral unit ball
Let Ω be the spectral unit ball of Mₙ(ℂ), that is, the set of n × n matrices with spectral radius less than 1. We are interested in classifying the automorphisms of Ω. We know that it is enough to consider the normalized automorphisms of Ω, that is, the automorphisms F satisfying F(0) = 0 and F'(0) = I, where I is the identity map on Mₙ(ℂ). The known normalized automorphisms are conjugations. Is every normalized automorphism a conjugation? We show that locally, in a neighborhood of a matrix with...