Let $D_j$ be a bounded hyperconvex domain in ${ℂ}^{n_j}$ and set $D=D₁\times \cdots \times D_s$, j=1,...,s, s≥ 3. Also let ₙ be the symmetrized polydisc in ℂⁿ, n ≥ 3. We characterize those real-valued continuous functions defined on the boundary of D or ₙ which can be extended to the inside to a pluriharmonic function. As an application a complete characterization of the compliant functions is obtained.

As a natural extension of bounded complete Reinhardt domains in ${ℂ}^N$ to spaces of continuous functions, continuous Reinhardt domains (CRD) are bounded open connected solid sets in commutative C*-algebras with respect to the natural ordering. We give a complete parametric description for the structure of holomorphic isomorphisms between CRDs and characterize the partial Jordan triple structures which can be associated with some CRDs. On the basis of these results, we test two conjectures concerning...

A differentiable group is a group in the category of (reduced and nonreduced) differentiable spaces. Special cases are the rationals ℚ, Lie groups, formal groups over ℝ or ℂ; in general there is some mixture of those types, the general structure, however, is not yet completely determined. The following gives as a corollary a first essential answer. It is shown, more generally,that a locally compact topological transformation group, operating effectively on a differentiable space X (which satisfies...

Nous montrons comment un cup-produit non trivial entre deux blocs de Jordan pour une même valeur propre de la monodromie agissant sur la cohomologie de la fibre de Milnor d’un germe de fonction holomorphe $f$ provoque des pôles d’ordres élevés pour le prolongement méromorphe de $|f{|}^{2z}$. Pour la valeur propre 1 ceci donne en particulier le phénomène de “contribution sur-effective”.

We discuss the existence of the current $g^{k}T$, $k\in \mathbb{N}$ for positive and closed currents $T$ and unbounded plurisubharmonic functions $g$. Furthermore, a new type of weighted Lelong number is introduced under the name of weight $k$ Lelong number.

Soit $E$, un fibré linéaire positif au-dessus d’une variété complexe compacte. Nous montrons que la fonction de distorsion définie par le rapport entre la métrique initiale et la métrique de Fubini-Study de $E^{\otimes k}$ admet un équivalent lorsque $k$ tend vers l’infini. Ceci améliore les encadrements de Kempf et Ji sur les variétés abéliennes, et les étend à toute variété projective. La démonstration repose sur le calcul d’un équivalent pour le noyau de la chaleur, avec contrôle de la convergence par rapport...

We study the relationship between convergence in capacities of plurisubharmonic functions and the convergence of the corresponding complex Monge-Ampère measures. We find one type of convergence of complex Monge-Ampère measures which is essentially equivalent to convergence in the capacity $C_n$ of functions. We also prove that weak convergence of complex Monge-Ampère measures is equivalent to convergence in the capacity $C_{n-1}$ of functions in some case. As applications we give certain stability theorems...

We prove that if $\left(\Omega \right)\ni u_j\to u\in \left(\Omega \right)$ in Cₙ-capacity then $limin{f}_{j\to \infty }{\left(d{d}^c{u}_j\right)}^n\ge 1_{u>-\infty }{\left(d{d}^{c}u\right)}^n$. This result is used to consider the convergence in capacity on bounded hyperconvex domains and compact Kähler manifolds.

We study restrictions of ω-plurisubharmonic functions to a smooth hypersurface S in a compact Kähler manifold X. The result obtained and the characterization of convergence in capacity due to S. Dinew and P. H. Hiep [to appear in Ann. Scuola Norm. Sup. Pisa Cl. Sci.] are used to study convergence in capacity on S.

We study the boundedness in $L^p{(}^n)$ of the projections onto spaces of functions with spectrum contained in horizontal strips. We obtain some results concerning convergence along nonisotropic regions of harmonic extensions of functions in $L^p{(}^n)$ with spectrum included in these horizontal strips.

The space $\mathscr{H}$ of Kähler metrics in a fixed Kähler class on a projective Kähler manifold $X$ is an infinite dimensional symmetric space whose geodesics ${\omega }_t$ are solutions of a homogeneous complex Monge-Ampère equation in $A\times X$, where $A\subset ℂ$ is an annulus. Phong-Sturm have proven that the Monge-Ampère geodesic of Kähler potentials $\varphi (t,z)$ of ${\omega }_t$ may be approximated in a weak $C^0$ sense by geodesics ${\varphi }_N(t,z)$ of the finite dimensional symmetric space of Bergman metrics of height $N$. In this article we prove that ${\varphi }_N(t,z)\to \varphi (t,z)$ in $C^2([0,1]\times X)$ in the case of...

We endow the module of analytic p-chains with the structure of a second-countable metrizable topological space.

In this paper we study the semigroups $\mathrm{\Phi }:{ℝ}^{+}\to Hol(D,D)$ of holomorphic maps of a strictly convex domain $D\subset {𝐂}^n$ into itself. In particular, we characterize the semigroups converging, uniformly on compact subsets, to a holomorphic map $h:D\to {𝐂}^n$.