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26. CJM 2006 (vol 58 pp. 262)
Connections on a Parabolic Principal Bundle Over a Curve The aim here is to define connections on a parabolic
principal bundle. Some applications are given.
Keywords:parabolic bundle, holomorphic connection, unitary connection Categories:53C07, 32L05, 14F05 
27. CJM 2005 (vol 57 pp. 506)
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 
28. CJM 2005 (vol 57 pp. 3)
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 $\mathcal{D}$.

29. CJM 2003 (vol 55 pp. 64)
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 
30. CJM 2002 (vol 54 pp. 1254)
Effective Actions of the Unitary Group on Complex Manifolds 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.
Keywords:complex manifolds, group actions Categories:32Q57, 32M17 
31. CJM 2002 (vol 54 pp. 493)
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 
32. CJM 2002 (vol 54 pp. 324)
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 
33. CJM 2002 (vol 54 pp. 55)
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 
34. CJM 2001 (vol 53 pp. 834)
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 
35. CJM 2001 (vol 53 pp. 73)
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 
36. CJM 2000 (vol 52 pp. 1149)
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 
37. CJM 2000 (vol 52 pp. 982)
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 
38. CJM 2000 (vol 52 pp. 1085)
Complex MongeAmpÃ¨re Measures of Plurisubharmonic Functions with Bounded Values Near the Boundary We give a characterization of bounded plurisubharmonic functions by
using their complex MongeAmp\`ere measures. This implies a both necessary
and sufficient condition for a positive measure to be complex
MongeAmp\`ere measure of some bounded plurisubharmonic function.
Categories:32F07, 32F05 
39. CJM 2000 (vol 52 pp. 348)
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 
40. CJM 2000 (vol 52 pp. 3)
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 
41. CJM 1999 (vol 51 pp. 915)
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 
42. CJM 1998 (vol 50 pp. 658)
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 
43. CJM 1998 (vol 50 pp. 99)
$A_\phi$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 
44. CJM 1997 (vol 49 pp. 1299)
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 
45. CJM 1997 (vol 49 pp. 1224)
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 
46. CJM 1997 (vol 49 pp. 916)
Quantization of the $4$dimensional nilpotent orbit of $\SL(3,\R)$ We give a new geometric model for the quantization
of the 4dimensional 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 nonflat Kaehler structure on
${\C}^2\{0\})$ (the universal cover of $\OR$) with Kaehler potential
$\rho=z^4$.
Categories:81S10, 32C17, 22E70 
47. CJM 1997 (vol 49 pp. 653)
On $\lowercase{q}$Carleson measures for spaces of ${\cal M}$harmonic functions 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
