Given a non-oscillating gradient trajectory $|\gamma|$ of a real analytic function $f$, we show that the limit $\nu$ of the secants at the limit point $\mathbf{0}$ of $|\gamma|$ along the trajectory $|\gamma|$ is an eigen-vector of the limit of the direction of the Hessian matrix $\operatorname{Hess} (f)$ at $\mathbf{0}$ along $|\gamma|$. The same holds true at infinity if the function is globally sub-analytic. We also deduce some interesting estimates along the trajectory. Away from the ends of the ambient space, this property is of metric nature and still holds in a general Riemannian analytic setting.

This article establishes a sufficient condition for Kobayashi hyperbolicity of unbounded domains in terms of curvature. Specifically, when $\Omega\subset{\mathbb C}^{n}$ corresponds to a sub-level set of a smooth, real-valued function $\Psi$, such that the form $\omega = {\bf i}\partial\bar{\partial}\Psi$ is Kähler and has bounded curvature outside a bounded subset, then this domain admits a hermitian metric of strictly negative holomorphic sectional curvature.

Given a real analytic set $X$ in a complex manifold and a positive integer $d$, denote by $\mathcal A^d$ the set of points $p$ in $X$ at which there exists a germ of a complex analytic set of dimension $d$ contained in $X$. It is proved that $\mathcal A^d$ is a closed semianalytic subset of $X$.

We revisit with another view point the construction by R. Brylinski and B. Kostant of minimal representations of simple Lie groups. We start from a pair $(V,Q)$, where $V$ is a complex vector space and $Q$ a homogeneous polynomial of degree 4 on $V$. The manifold $\Xi $ is an orbit of a covering of ${\rm Conf}(V,Q)$, the conformal group of the pair $(V,Q)$, in a finite dimensional representation space. By a generalized Kantor-Koecher-Tits construction we obtain a complex simple Lie algebra $\mathfrak g$, and furthermore a real form ${\mathfrak g}_{\mathbb R}$. The connected and simply connected Lie group $G_{\mathbb R}$ with ${\rm Lie}(G_{\mathbb R})={\mathfrak g}_{\mathbb R}$ acts unitarily on a Hilbert space of holomorphic functions defined on the manifold $\Xi $.

We study properties of composition operators induced by symbols acting from the unit disk to the polydisk. This result will be involved in the investigation of weighted composition operators on the Hardy space on the unit disk and moreover be concerned with composition operators acting from the Bergman space to the Hardy space on the unit disk.

An extension theorem for holomorphic mappings between two domains in $\mathbb C^2$ is proved under purely local hypotheses.

If $\Phi_\lambda$ is a master function corresponding to a hyperplane arrangement $\mathcal A$ and a collection of weights $\lambda$, we investigate the relationship between the critical set of $\Phi_\lambda$, the variety defined by the vanishing of the one-form $\omega_\lambda=\operatorname{d} \log \Phi_\lambda$, and the resonance of $\lambda$. For arrangements satisfying certain conditions, we show that if $\lambda$ is resonant in dimension $p$, then the critical set of $\Phi_\lambda$ has codimension at most $p$. These include all free arrangements and all rank $3$ arrangements.

The moduli space of smooth real binary octics has five connected components. They parametrize the real binary octics whose defining equations have $0,\dots,4$ complex-conjugate pairs of roots respectively. We show that each of these five components has a real hyperbolic structure in the sense that each is isomorphic as a real-analytic manifold to the quotient of an open dense subset of $5$-dimensional real hyperbolic space $\mathbb{RH}^5$ by the action of an arithmetic subgroup of $\operatorname{Isom}(\mathbb{RH}^5)$. These subgroups are commensurable to discrete hyperbolic reflection groups, and the Vinberg diagrams of the latter are computed.

Let $\mathcal{D}$ be the $n$-dimensional Lie ball and let $\mathbf{B}(S)$ be the space of hyperfunctions on the Shilov boundary $S$ of $\mathcal{D}$. The aim of this paper is to give a necessary and sufficient condition on the generalized Poisson transform $P_{l,\lambda}f$ of an element $f$ in the space $\mathbf{B}(S)$ for $f$ to be in $ L^{p}(S)$, $1 > p > \infty.$ Namely, if $F$ is the Poisson transform of some $f\in \mathbf{B}(S)$ (i.e., $F=P_{l,\lambda}f$), then for any $l\in \mathbb{Z}$ and $\lambda\in \mathbb{C}$ such that $\mathcal{R}e[i\lambda] > \frac{n}{2}-1$, we show that $f\in L^{p}(S)$ if and only if $f$ satisfies the growth condition $$ \|F\|_{\lambda,p}=\sup_{0\leq r < 1}(1-r^{2})^{\mathcal{R}e[i\lambda]-\frac{n}{2}+l}\Big[\int_{S}|F(ru)|^{p}du \Big]^{\frac{1}{p}} < +\infty. $$

Let ${\mathcal T}$ be the $C^\ast $-algebra generated by the Toeplitz operators $\{T_\varphi : \varphi \in L^\infty (S,d\sigma )\}$ on the Hardy space $H^2(S)$ of the unit sphere in $\mathbf{C}^n$. It is well known that ${\mathcal T}$ is contained in the essential commutant of $\{T_\varphi : \varphi \in \operatorname{VMO}\cap L^\infty (S,d\sigma )\}$. We show that the essential commutant of $\{T_\varphi : \varphi \in \operatorname{VMO}\cap L^\infty (S,d\sigma )\}$ is strictly larger than ${\mathcal T}$.

Let $( X,g) $ be a complete noncompact Kähler manifold, of dimension $n\geq2,$ with positive Ricci curvature and of standard type (see the definition below). N. Mok proved that $X$ can be compactified, \emph{i.e.,} $X$ is biholomorphic to a quasi-projective variety$.$ The aim of this paper is to prove that the $L^{2}$ holomorphic sections of the line bundle $K_{X}^{-q}$ and the volume form of the metric $g$ have no essential singularities near the divisor at infinity. As a consequence we obtain a comparison between the volume forms of the Kähler metric $g$ and of the Fubini--Study metric induced on $X$. In the case of $\dim_{\mathbb{C} }X=2,$ we establish a relation between the number of components of the divisor $D$ and the dimension of the groups $H^{i}( \overline{X}, \Omega_{\overline{X}}^{1}( \log D) )$.

In this paper we study bilinear Hankel forms of higher weights on Hardy spaces in several dimensions. (The Schatten class Hankel forms of higher weights on weighted Bergman spaces have already been studied by Janson and Peetre for one dimension and by Sundhäll for several dimensions). We get a full characterization of Schatten class Hankel forms in terms of conditions for the symbols to be in certain Besov spaces. Also, the Hankel forms are bounded and compact if and only if the symbols satisfy certain Carleson measure criteria and vanishing Carleson measure criteria, respectively.

We introduce a wide subclass ${\mathcal F}(X,\omega)$ of quasi-plurisubharmonic functions in a compact Kähler manifold, on which the complex Monge-Ampère operator is well defined and the convergence theorem is valid. We also prove that ${\mathcal F}(X,\omega)$ is a convex cone and includes all quasi-plurisubharmonic functions that are in the Cegrell class.

In this work, we investigate how to decompose a pair $(A,B)$ of loxodromic isometries of the complex hyperbolic plane $\mathbf H^{2}_{\mathbb C}$ under the form $A=I_1I_2$ and $B=I_3I_2$, where the $I_k$'s are involutions. The main result is a decomposability criterion, which is expressed in terms of traces of elements of the group $\langle A,B\rangle$.

In this paper we study the notion of a convex subordination chain in several complex variables. We obtain certain necessary and sufficient conditions for a mapping to be a convex subordination chain, and we give various examples of convex subordination chains on the Euclidean unit ball in $\mathbb{C}^n$. We also obtain a sufficient condition for injectivity of $f(z/\|z\|,\|z\|)$ on $B^n\setminus\{0\}$, where $f(z,t)$ is a convex subordination chain over $(0,1)$.

Given a positive continuous function $\mu$ on the interval $0
Categories:47B33, 32A70, 46E15

This paper deals with the extension of CR functions from a manifold $M\subset \mathbb C^n$ into directions produced by higher order commutators of holomorphic and antiholomorphic vector fields. It uses the theory of complex ``sectors'' attached to real submanifolds introduced in recent joint work of the authors with D. Zaitsev. In addition, it develops a new technique of approximation of sectors by smooth discs.

We obtain a uniform linear bound for the Chevalley function at a point in the source of an analytic mapping that is regular in the sense of Gabrielov. There is a version of Chevalley's lemma also along a fibre, or at a point of the image of a proper analytic mapping. We get a uniform linear bound for the Chevalley function of a closed Nash (or formally Nash) subanalytic set.

Let~$V$ be an analytic variety in some open set in~$\C^n$. For a real analytic curve~$\gamma$ with $ \gamma(0) = 0 $ and $ d \ge 1 $ define $ V_t = t^{-d}(V - \gamma(t)) $. It was shown in a previous paper that the currents of integration over~$V_t$ converge to a limit current whose support $ T_{\gamma,d} V $ is an algebraic variety as~$t$ tends to zero. Here, it is shown that the canonical defining function of the limit current is the suitably normalized limit of the canonical defining functions of the~$V_t$. As a corollary, it is shown that $ T_{\gamma,d} V $ is either inhomogeneous or coincides with $ T_{\gamma,\delta} V $ for all~$\delta$ in some neighborhood of~$d$. As another application it is shown that for surfaces only a finite number of curves lead to limit varieties that are interesting for the investigation of Phragm\'en--Lindel\"of conditions. Corresponding results for limit varieties $ T_{\sigma,\delta} W $ of algebraic varieties W along real analytic curves tending to infinity are derived by a reduction to the local case.

Meromorphic continuation of the Eisenstein series induced from spherical, cuspidal data on parabolic subgroups is achieved via reworking Bernstein's adaptation of Selberg's third proof of meromorphic continuation.

In this paper we study ruled surfaces which appear as an exceptional surface in a succession of blowing-ups. In particular we prove that the $e$-invariant of such a ruled exceptional surface $E$ is strictly positive whenever its intersection with the other exceptional surfaces does not contain a fiber (of $E$). This fact immediately enables us to resolve an open problem concerning an intersection configuration on such a ruled exceptional surface consisting of three nonintersecting sections. In the second part of the paper we apply the non-vanishing of $e$ to the study of the poles of the well-known topological, Hodge and motivic zeta functions.

Le diagramme d'Eisenbud et Neumann d'un germe est un arbre qui repr\'esente ce germe et permet d'en calculer les invariants. On donne une d\'emonstration alg\'ebrique d'un r\'esultat caract\'erisant l'ensemble des quotients jacobiens d'un germe d'application $(f,g)$ \`a partir du diagramme d'Eisenbud et Neumann de $fg$.

We study complex commutative Banach algebras (and, more generally, continuous inverse algebras) in which the holomorphic functions of a fixed $n$-tuple of elements are dense. In particular, we characterize the compact subsets of~$\C^n$ which appear as joint spectra of such $n$-tuples. The characterization is compared with several established notions of holomorphic convexity by means of approximation conditions.

The aim here is to define connections on a parabolic principal bundle. Some applications are given.

Contractivity and hypercontractivity properties of semigroups are now well understood when the generator, $A$, is a Dirichlet form operator. It has been shown that in some holomorphic function spaces the semigroup operators, $e^{-tA}$, can be bounded {\it below} from $L^p$ to $L^q$ when $p,q$ and $t$ are suitably related. We will show that such lower boundedness occurs also in spaces of subharmonic functions.