Traces of Runge domians on analytic subsets.
Transfinite diameter, Chebyshev constants, and capacities in ℂⁿ
The famous result of geometric complex analysis, due to Fekete and Szegö, states that the transfinite diameter d(K), characterizing the asymptotic size of K, the Chebyshev constant τ(K), characterizing the minimal uniform deviation of a monic polynomial on K, and the capacity c(K), describing the asymptotic behavior of the Green function at infinity, coincide. In this paper we give a survey of results on multidimensional notions of transfinite diameter, Chebyshev constants and capacities, related...
Transfinite diameter on curves, Hankel determinants and the moment problem
We discuss problems on Hankel determinants and the classical moment problem related to and inspired by certain Vandermonde determinants for polynomial interpolation on (quadratic) algebraic curves in ℂ².
Tubenumgebungen Steinscher Räume.
Two remarks on the Suita conjecture
It is shown that the weak multidimensional Suita conjecture fails for any bounded non-pseudoconvex domain with -smooth boundary. On the other hand, it is proved that the weak converse to the Suita conjecture holds for any finitely connected planar domain.
Two Theorems on Extensions of Holomorphic Mappings.
Un exemple de fibré holomorphe non de Stein a fibre C2 ayant pour base le disque ou le plan.
Un théorème de Bloch presque complexe
Cet article est consacré à la démonstration d’une version presque complexe du théorème de Bloch. Considérons la réunion C de quatre J-droites en position générale dans un plan projectif presque complexe. Nous démontrons que toute suite non normale de J-disques évitant évitant la configuration C admet une sous-suite convergeant, au sens de Hausdorff, vers une partie la réunion des diagonales de C. En particulier, le complémentaire de la configuration C est hyperboliquement plongé dans le paln projectif...
Une caractérisation géométrique des exemples de Lattès de
Un exemple de Lattès est un endomorphisme holomorphe de l’espace projectif complexe qui se relève en une dilatation de l’espace affine de même dimension au moyen d’un revêtement ramifié sur les fibres duquel un groupe cristallographique agit transitivement. Nous montrons que tout endomorphisme holomorphe d’un espace projectif complexe dont le courant de Green est lisse et strictement positif sur un ouvert non vide est nécessairement un exemple de Lattès.
Une estimation des coefficients tangents d'un courant positif fermé dans un domaine de C3.
In this paper, we study the behaviour near the boundary of the complex tangent coefficients of a closed positive current in a bounded domain of C3 with C∞ boundary. Assuming that the current satisfies the Blaschke condition, we give a condition on the complex tangent coefficients which is better than the one which can be proved using the pseudo-distance introduced by A. Nagel, E. Stein and S. Wainger (in analogy with the case of domains in C2). Moreover, when the domain is supposed to be pseudoconvex,...
Une nouvelle version du théorème d'extension de Hartogs pour les applications séparément holomorphes entre espaces analytiques
This paper is concerned with the problem of extension of separately holomorphic mappings defined on a "generalized cross" of a product of complex analytic spaces with values in a complex analytic space. The crosses considered here are inscribed in Borel rectangles (of a product of two complex analytic spaces) which are not necessarily open but are non-pluripolar and can be quite small from the topological point of view. Our first main result says that the singular...
Uniqueness and factorization of Coleff-Herrera currents
We prove a uniqueness result for Coleff-Herrera currents which in particular means that if defines a complete intersection, then the classical Coleff-Herrera product associated to is the unique Coleff-Herrera current that is cohomologous to with respect to the operator , where is interior multiplication with . From the uniqueness result we deduce that any Coleff-Herrera current on a variety is a finite sum of products of residue currents with support on and holomorphic forms.
Uniqueness of equivariant singular Bott-Chern classes
In this paper, we shall discuss possible theories of defining equivariant singular Bott-Chern classes and corresponding uniqueness property. By adding a natural axiomatic characterization to the usual ones of equivariant Bott-Chern secondary characteristic classes, we will see that the construction of Bismut’s equivariant Bott-Chern singular currents provides a unique way to define a theory of equivariant singular Bott-Chern classes. This generalizes J. I. Burgos Gil and R. Liţcanu’s discussion...
Weak solutions of equations of complex Monge-Ampère type
We prove some existence results for equations of complex Monge-Ampère type in strictly pseudoconvex domains and on Kähler manifolds.
Weak solutions to the complex Hessian equation
We investigate the class of functions associated with the complex Hessian equation .
Weak solutions to the complex Monge-Ampère equation on hyperconvex domains
We show a very general existence theorem for a complex Monge-Ampère type equation on hyperconvex domains.
Weighted Bernstein-Markov property in ℂⁿ
We study the weighted Bernstein-Markov property for subsets in ℂⁿ which might not be bounded. An application concerning approximation of the weighted Green function using Bergman kernels is also given.
Weighted pluripotential theory on compact Kähler manifolds
We introduce a weighted version of the pluripotential theory on compact Kähler manifolds developed by Guedj and Zeriahi. We give the appropriate definition of a weighted pluricomplex Green function, its basic properties and consider its behavior under holomorphic maps. We also develop a homogeneous version of the weighted theory and establish a generalization of Siciak's H-principle.
Weighted θ-incomplete pluripotential theory
Weighted pluripotential theory is a rapidly developing area; and Callaghan [Ann. Polon. Math. 90 (2007)] recently introduced θ-incomplete polynomials in ℂ for n>1. In this paper we combine these two theories by defining weighted θ-incomplete pluripotential theory. We define weighted θ-incomplete extremal functions and obtain a Siciak-Zahariuta type equality in terms of θ-incomplete polynomials. Finally we prove that the extremal functions can be recovered using orthonormal polynomials and we...
Whitney's extension theorem for nonquasianalytic classes of ultradifferentiable functions