We study adjacency of equisingularity types of planar complex curve singularities in terms of their Enriques diagrams. The goal is, given two equisingularity types, to determine whether one of them is adjacent to the other. For linear adjacency a complete answer is obtained, whereas for arbitrary (analytic) adjacency a necessary condition and a sufficient condition are proved. We also obtain new examples of exceptional deformations, {\em i.e.,} singular curves of type $\mathcal{D}'$ that can be deformed to a curve of type $\mathcal{D}$ without $\mathcal{D}'$ being adjacent to $\mathcal{D}$.

Let $V$ be an analytic variety in some open set in $\mathbb{C}^n$ which contains the origin and which is purely $k$-dimensional. For a curve $\gamma$ in $\mathbb{C}^n$, defined by a convergent Puiseux series and satisfying $\gamma(0) = 0$, and $d \ge 1$, define $V_t := t^{-d} \bigl( V-\gamma(t) \bigr)$. Then the currents defined by $V_t$ converge to a limit current $T_{\gamma,d} [V]$ as $t$ tends to zero. $T_{\gamma,d} [V]$ is either zero or its support is an algebraic variety of pure dimension $k$ in $\mathbb{C}^n$. Properties of such limit currents and examples are presented. These results will be applied in a forthcoming paper to derive necessary conditions for varieties satisfying the local Phragm\'en-Lindel\"of condition that was used by H\"ormander to characterize the constant coefficient partial differential operators which act surjectively on the space of all real analytic functions on $\mathbb{R}^n$.

We classify all connected $n$-dimensional complex manifolds admitting effective actions of the unitary group $U_n$ by biholomorphic transformations. One consequence of this classification is a characterization of $\CC^n$ by its automorphism group.

We compute the category of perverse sheaves on Hermitian symmetric spaces in types~A and D, constructible with respect to the Schubert stratification. The calculation is microlocal, and uses the action of the Borel group to study the geometry of the conormal variety $\Lambda$.

Let $B$ be the unit ball of $\bb{C}^n$ with respect to an arbitrary norm. We prove that the analog of the Carath\'eodory set, {\it i.e.} the set of normalized holomorphic mappings from $B$ into $\bb{C}^n$ of ``positive real part'', is compact. This leads to improvements in the existence theorems for the Loewner differential equation in several complex variables. We investigate a subset of the normalized biholomorphic mappings of $B$ which arises in the study of the Loewner equation, namely the set $S^0(B)$ of mappings which have parametric representation. For the case of the unit polydisc these mappings were studied by Poreda, and on the Euclidean unit ball they were studied by Kohr. As in Kohr's work, we consider subsets of $S^0(B)$ obtained by placing restrictions on the mapping from the Carath\'eodory set which occurs in the Loewner equation. We obtain growth and covering theorems for these subsets of $S^0(B)$ as well as coefficient estimates, and consider various examples. Also we shall see that in higher dimensions there exist mappings in $S(B)$ which can be imbedded in Loewner chains, but which do not have parametric representation.

We verify a generalization of (3.3) from \cite{Le} proving that the homotopy type of the Milnor fiber of a reduced hypersurface singularity depends only on the embedded topological type of the singularity. In particular, using \cite{Za,Li1,Oh1,Gau} for irreducible quasi-ordinary germs, it depends only on the normalized distinguished pairs of the singularity. The main result of the paper provides an explicit formula for the Euler-characteristic of the Milnor fiber in the surface case.

Let $X$ be a nonsingular algebraic variety in characteristic zero. To an effective divisor on $X$ Kontsevich has associated a certain motivic integral, living in a completion of the Grothendieck ring of algebraic varieties. He used this invariant to show that birational (smooth, projective) Calabi-Yau varieties have the same Hodge numbers. Then Denef and Loeser introduced the invariant {\it motivic (Igusa) zeta function}, associated to a regular function on $X$, which specializes to both the classical $p$-adic Igusa zeta function and the topological zeta function, and also to Kontsevich's invariant. This paper treats a generalization to singular varieties. Batyrev already considered such a `Kontsevich invariant' for log terminal varieties (on the level of Hodge polynomials of varieties instead of in the Grothendieck ring), and previously we introduced a motivic zeta function on normal surface germs. Here on any $\bbQ$-Gorenstein variety $X$ we associate a motivic zeta function and a `Kontsevich invariant' to effective $\bbQ$-Cartier divisors on $X$ whose support contains the singular locus of~$X$.

In this paper, we investigate stratification theory in terms of the defining equations of strata and maps (without tube systems), offering a concrete approach to show that some given family is topologically trivial. In this approach, we consider a weighted version of $(w)$-regularity condition and Kuo's ratio test condition.

We describe the embedded resolution of an irreducible quasi-ordinary surface singularity $(V,p)$ which results from applying the canonical resolution of Bierstone-Milman to $(V,p)$. We show that this process depends solely on the characteristic pairs of $(V,p)$, as predicted by Lipman. We describe the process explicitly enough that a resolution graph for $f$ could in principle be obtained by computer using only the characteristic pairs.

Let $Y$ be an infinite covering space of a projective manifold $M$ in $\P^N$ of dimension $n\geq 2$. Let $C$ be the intersection with $M$ of at most $n-1$ generic hypersurfaces of degree $d$ in $\mathbb{P}^N$. The preimage $X$ of $C$ in $Y$ is a connected submanifold. Let $\phi$ be the smoothed distance from a fixed point in $Y$ in a metric pulled up from $M$. Let $\O_\phi(X)$ be the Hilbert space of holomorphic functions $f$ on $X$ such that $f^2 e^{-\phi}$ is integrable on $X$, and define $\O_\phi(Y)$ similarly. Our main result is that (under more general hypotheses than described here) the restriction $\O_\phi(Y) \to \O_\phi(X)$ is an isomorphism for $d$ large enough. This yields new examples of Riemann surfaces and domains of holomorphy in $\C^n$ with corona. We consider the important special case when $Y$ is the unit ball $\B$ in $\C^n$, and show that for $d$ large enough, every bounded holomorphic function on $X$ extends to a unique function in the intersection of all the nontrivial weighted Bergman spaces on $\B$. Finally, assuming that the covering group is arithmetic, we establish three dichotomies concerning the extension of bounded holomorphic and harmonic functions from $X$ to $\B$.

We give a characterization of bounded plurisubharmonic functions by using their complex Monge-Amp\`ere measures. This implies a both necessary and sufficient condition for a positive measure to be complex Monge-Amp\`ere measure of some bounded plurisubharmonic function.

Nous \'etudions les polyn\^omes $F \in \C \{S_\tau\} [Y] $ \`a coefficients dans l'anneau de germes de fonctions holomorphes au point sp\'ecial d'une vari\'et\'e torique affine. Nous g\'en\'eralisons \`a ce cas la param\'etrisation classique des singularit\'es quasi-ordinaires. Cela fait intervenir d'une part une g\'en\'eralization de l'algorithme de Newton-Puiseux, et d'autre part une relation entre le poly\`edre de Newton du discriminant de $F$ par rapport \`a $Y$ et celui de $F$ au moyen du polytope-fibre de Billera et Sturmfels~\cite{Sturmfels}. Cela nous permet enfin de calculer, sous des hypoth\`eses de non d\'eg\'en\'erescence, les sommets du poly\`edre de Newton du discriminant a partir de celui de $F$, et les coefficients correspondants \`a partir des coefficients des exposants de $F$ qui sont dans les ar\^etes de son poly\`edre de Newton.

Using Local Residues and the Duality Principle a multidimensional variation of the completeness theorems by T.~Carleman and A.~F.~Leontiev is proven for the space of holomorphic functions defined on a suitable open strip $T_{\alpha}\subset {\bf C}^2$. The completeness theorem is a direct consequence of the Cauchy Residue Theorem in a torus. With suitable modifications the same result holds in ${\bf C}^n$.

Consider the polynomial hull of a smoothly varying family of strictly convex smooth domains fibered over the unit circle. It is well-known that the boundary of the hull is foliated by graphs of analytic discs. We prove that this foliation is smooth, and we show that it induces a complex flow of contactomorphisms. These mappings are quasiconformal in the sense of Kor\'anyi and Reimann. A similar bound on their quasiconformal distortion holds as in the one-dimensional case of holomorphic motions. The special case when the fibers are rotations of a fixed domain in $\C^2$ is studied in details.

The aim of this paper is to study small Hankel operators $h$ on the Hardy space or on weighted Bergman spaces, where $\Omega$ is a finite type domain in ${\Bbbvii C}^2$ or a strictly pseudoconvex domain in ${\Bbbvii C}^n$. We give a sufficient condition on the symbol $f$ so that $h$ belongs to the Schatten class ${\cal S}_p$, $1\le p<+\infty$.

Generalizing the notion of invariant subspaces on the 2-dimensional torus $T^2$, we study the structure of $A_\phi$-invariant subspaces of $L^2(T^2)$. A complete description is given of $A_\phi$-invariant subspaces that satisfy conditions similar to those studied by Mandrekar, Nakazi, and Takahashi.

In this paper, we solve the $\dbar$-Neumann problem on $(0,q)$ forms, $0\leq q \leq n$, in the strictly pseudoconvex non-isotropic Siegel domain: \[ \cU=\left\{ \begin{array}{clc} &\bz=(z_1,\ldots,z_n) \in \C^{n},\\ (\bz,z_{n+1}):&&\Im (z_{n+1}) > \sum_{j=1}^{n}a_j |z_j|^2 \\ &z_{n+1}\in \C; \end{array} \right\}, \] where $a_j> 0$ for $j=1,2,\ldots, n$. The metric we use is invariant under the action of the Heisenberg group on the domain. The fundamental solution of the related differential equation is derived via the Laguerre calculus. We obtain an explicit formula for the kernel of the Neumann operator. We also construct the solution of the corresponding heat equation and the fundamental solution of the Laplacian operator on the Heisenberg group.

We give the irreducible decomposition of the tensor product of an analytic continuation of the holomorphic discrete series of $\SU(2, 2)$ with its conjugate.

We give a new geometric model for the quantization of the 4-dimensional conical (nilpotent) adjoint orbit $\OR$ of $\SL(3,\R)$. The space of quantization is the space of holomorphic functions on ${\C}^2-\{0\})$ which are square integrable with respect to a signed measure defined by a Meijer $G$-function. We construct the quantization out a non-flat Kaehler structure on ${\C}^2-\{0\})$ (the universal cover of $\OR$) with Kaehler potential $\rho=|z|^4$.

In this paper we study the $q$-Carleson measures for a space $h_\alpha^p$ of ${\cal M}$-harmonic potentials in the unit ball of $\C^n$, when $q
Categories:32A35, 31C15