26. CJM 2007 (vol 59 pp. 3)
 Biller, Harald

Holomorphic Generation of Continuous Inverse Algebras
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.
Keywords:holomorphic functional calculus, commutative continuous inverse algebra, holomorphic convexity, Stein compacta, meromorphic convexity, holomorphic approximation Categories:46H30, 32A38, 32E30, 41A20, 46J15 

27. CJM 2006 (vol 58 pp. 262)
28. CJM 2005 (vol 57 pp. 506)
 Gross, Leonard; Grothaus, Martin

Reverse Hypercontractivity for Subharmonic Functions
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.
Keywords:Reverse hypercontractivity, subharmonic Categories:58J35, 47D03, 47D07, 32Q99, 60J35 

29. CJM 2005 (vol 57 pp. 3)
 AlberichCarramiñana, Maria; Roé, Joaquim

Enriques Diagrams and Adjacency of Planar Curve Singularities
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 $\mathcal{D}$.


30. CJM 2003 (vol 55 pp. 64)
 Braun, Rüdiger W.; Meise, Reinhold; Taylor, B. A.

Higher Order Tangents to Analytic Varieties along Curves
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\'enLindel\"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 $\mathbb{R}^n$.
Category:32C25 

31. CJM 2002 (vol 54 pp. 1254)
32. CJM 2002 (vol 54 pp. 493)
 Braden, Tom

Perverse Sheaves on Grassmannians
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
$\Lambda$.
Keywords:perverse sheaves, microlocal geometry Categories:32S60, 32C38, 35A27 

33. CJM 2002 (vol 54 pp. 324)
 Graham, Ian; Hamada, Hidetaka; Kohr, Gabriela

Parametric Representation of Univalent Mappings in Several Complex Variables
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.
Categories:32H02, 30C45 

34. CJM 2002 (vol 54 pp. 55)
 Ban, Chunsheng; McEwan, Lee J.; Némethi, András

On the Milnor Fiber of a Quasiordinary Surface Singularity
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 quasiordinary germs,
it depends only on the normalized distinguished pairs of the
singularity. The main result of the paper provides an explicit
formula for the Eulercharacteristic of the Milnor fiber in the
surface case.
Categories:14B05, 14E15, 32S55 

35. CJM 2001 (vol 53 pp. 834)
 Veys, Willem

Zeta Functions and `Kontsevich Invariants' on Singular Varieties
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) CalabiYau 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~$X$.
Keywords:singularity invariant, topological zeta function, motivic zeta function Categories:14B05, 14E15, 32S50, 32S45 

36. CJM 2001 (vol 53 pp. 73)
 Fukui, Toshizumi; Paunescu, Laurentiu

Stratification Theory from the Weighted Point of View
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.
Categories:32B99, 14P25, 32Cxx, 58A35 

37. CJM 2000 (vol 52 pp. 1149)
 Ban, Chunsheng; McEwan, Lee J.

Canonical Resolution of a Quasiordinary Surface Singularity
We describe the embedded resolution of an irreducible quasiordinary
surface singularity $(V,p)$ which results from applying the canonical
resolution of BierstoneMilman 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.
Keywords:canonical resolution, quasiordinary singularity Categories:14B05, 14J17, 32S05, 32S25 

38. CJM 2000 (vol 52 pp. 982)
 Lárusson, Finnur

Holomorphic Functions of Slow Growth on Nested Covering Spaces of Compact Manifolds
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 $n1$ 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 $\B$.
Categories:32A10, 14E20, 30F99, 32M15 

39. CJM 2000 (vol 52 pp. 1085)
40. CJM 2000 (vol 52 pp. 348)
 González Pérez, P. D.

SingularitÃ©s quasiordinaires toriques et polyÃ¨dre de Newton du discriminant
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 quasiordinaires. Cela fait intervenir d'une part
une g\'en\'eralization de l'algorithme de NewtonPuiseux, 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
polytopefibre 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.
Categories:14M25, 32S25 

41. CJM 2000 (vol 52 pp. 3)
 Aizenberg, Lev; Vidras, Alekos

On Small Complete Sets of Functions
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 ${\bf C}^n$.
Categories:32A10, 42C30 

42. CJM 1999 (vol 51 pp. 915)
 Balogh, Zoltán M.; Leuenberger, Christoph

Quasiconformal Contactomorphisms and Polynomial Hulls with Convex Fibers
Consider the polynomial hull of a smoothly varying family of
strictly convex smooth domains fibered over the unit circle. It is
wellknown 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
onedimensional case of holomorphic motions. The special case when
the fibers are rotations of a fixed domain in $\C^2$ is studied in
details.
Categories:32E20, 30C65 

43. CJM 1998 (vol 50 pp. 658)
 Symesak, Frédéric

Hankel operators on pseudoconvex domains of finite type in ${\Bbb C}^2$
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 ${\cal S}_p$,
$1\le p<+\infty$.
Categories:32A37, 47B35, 47B10, 46E22 

44. CJM 1998 (vol 50 pp. 99)
 Izuchi, Keiji; Matsugu, Yasuo

$A_\phi$invariant subspaces on the torus
Generalizing the notion of invariant subspaces on
the 2dimensional 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.
Categories:32A35, 47A15 

45. CJM 1997 (vol 49 pp. 1299)
 Tie, Jingzhi

The explicit solution of the $\bar\partial$Neumann problem in a nonisotropic Siegel domain
In this paper, we solve the $\dbar$Neumann problem
on $(0,q)$ forms, $0\leq q \leq n$, in the strictly
pseudoconvex nonisotropic 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.
Categories:32F15, 32F20, 35N15 

46. CJM 1997 (vol 49 pp. 1224)
 Ørsted, Bent; Zhang, Genkai

Tensor products of analytic continuations of holomorphic discrete series
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.
Keywords:Holomorphic discrete series, tensor product, spherical function, ClebschGordan coefficient, Plancherel theorem Categories:22E45, 43A85, 32M15, 33A65 

47. CJM 1997 (vol 49 pp. 916)
 Brylinski, Ranee

Quantization of the $4$dimensional nilpotent orbit of SL(3,$\mathbb{R}$)
We give a new geometric model for the quantization
of the 4dimensional conical (nilpotent) adjoint orbit
$O_\mathbb{R}$ of SL$(3,\mathbb{R})$. The space of quantization is the space of
holomorphic functions on $\mathbb{C}^2 \{ 0 \}$ which are square integrable
with respect to a signed measure defined by a Meijer $G$function.
We construct the quantization out a nonflat Kaehler structure on
$\mathbb{C}^2  \{ 0 \}$ (the universal cover of $O_\mathbb{R}$ ) with Kaehler potential
$\rho=z^4$.
Categories:81S10, 32C17, 22E70 

48. CJM 1997 (vol 49 pp. 653)