76. CMB 2000 (vol 43 pp. 174)
77. CMB 2000 (vol 43 pp. 47)
 Božičević, Mladen

A Property of Lie Group Orbits
Let $G$ be a real Lie group and $X$ a real analytic manifold.
Suppose that $G$ acts analytically on $X$ with finitely many
orbits. Then the orbits are subanalytic in $X$. As a consequence
we show that the microsupport of a $G$equivariant sheaf on $X$ is
contained in the conormal variety of the $G$action.
Categories:32B20, 22E15 

78. CMB 1999 (vol 42 pp. 499)
 Zaharia, Alexandru

Characterizations of Simple Isolated Line Singularities
A line singularity is a function germ $f\colon(\CC ^{n+1},0) \lra\CC$
with a smooth $1$dimensional critical set $\Sigma=\{(x,y)\in \CC\times
\CC^n \mid y=0\}$. An isolated line singularity is defined by the
condition that for every $x \neq 0$, the germ of $f$ at $(x,0)$ is
equivalent to $y_1^2 +\cdots+y_n ^2$. Simple isolated line
singularities were classified by Dirk Siersma and are analogous
of the famous $ADE$ singularities. We give two new
characterizations of simple isolated line singularities.
Categories:32S25, 14B05 

79. CMB 1999 (vol 42 pp. 97)
 Kwon, E. G.

On Analytic Functions of Bergman $\BMO$ in the Ball
Let $B = B_n$ be the open unit ball of $\bbd C^n$ with
volume measure $\nu$, $U = B_1$ and ${\cal B}$ be the Bloch space on
$U$. ${\cal A}^{2, \alpha} (B)$, $1 \leq \alpha < \infty$, is defined
as the set of holomorphic $f\colon B \rightarrow \bbd C$ for which
$$
\int_B \vert f(z) \vert^2 \left( \frac 1{\vert z\vert}
\log \frac 1{1  \vert z\vert } \right)^{\alpha}
\frac {d\nu (z)}{1\vert z\vert} < \infty
$$
if $0 < \alpha <\infty$ and ${\cal A}^{2, 1} (B) = H^2(B)$, the Hardy
space. Our objective of this note is to characterize, in terms of
the Bergman distance, those holomorphic $f\colon B \rightarrow U$ for
which the composition operator $C_f \colon {\cal B} \rightarrow
{\cal A}^{2, \alpha}(B)$ defined by $C_f (g) = g\circ f$,
$g \in {\cal B}$, is bounded. Our result has a corollary that
characterize the set of analytic functions of bounded mean
oscillation with respect to the Bergman metric.
Keywords:Bergman distance, \BMOA$, Hardy space, Bloch function Category:32A37 

80. CMB 1998 (vol 41 pp. 129)
81. CMB 1997 (vol 40 pp. 356)
 Mazet, Pierre

Principe du maximum et lemme de Schwarz, a valeurs vectorielles
Nous {\'e}tablissons un
th{\'e}or{\`e}me pour les fonctions holomorphes {\`a} valeurs dans une
partie convexe ferm{\'e}e. Ce th{\'e}or{\`e}me pr{\'e}cise
la position des coefficients de Taylor de telles fonctions et peut
{\^e}tre consid{\'e}r{\'e} comme une g{\'e}n{\'e}ralisation des
in{\'e}galit{\'e}s de Cauchy. Nous montrons alors comment ce
th{\'e}or{\`e}me permet de retrouver des versions connues du principe
du maximum et d'obtenir de nouveaux r{\'e}sultats sur les
applications holomorphes {\`a} valeurs vectorielles.
Keywords:Principe du maximum, lemme de Schwarz, points extr{Ã©maux. Categories:30C80, 32A30, 46G20, 52A07 

82. CMB 1997 (vol 40 pp. 129)
83. CMB 1997 (vol 40 pp. 117)
 Vigué, JeanPierre

Un lemme de Schwarz pour les boulesunitÃ©s ouvertes
Let $B_1$ and $B_2$ be the open unit balls of ${\bbd C}^{n_1}$ and
${\bbd C}^{n_2}$ for the norms $\Vert\,{.}\,\Vert_1$ and $\Vert\,{.}\,
\Vert_2$. Let $f \colon B_1 \rightarrow B_2$ be a holomorphic
mapping such that $f(0)=0$. It is well known that, for every $z \in B_1$,
$\Vert f(z)\Vert_2 \leq \Vert z \Vert_1$, and $\Vert f'(0)\Vert \leq
1$.
In this paper, I prove the converse of this result. Let $f \colon B_1
\rightarrow B_2$ be a holomorphic mapping such that $f'(0)$ is an
isometry. If $B_2$ is strictly convex, I prove that $f(0) =0$ and
that $f$ is linear. I also define the rank of a point $x$ belonging to the
boundary of $B_1$ or $B_2$. Under some hypotheses on the ranks, I
prove that a holomorphic mapping such that $f(0) = 0$ and that $f'(0)$ is
an isometry is linear.
Categories:32H15, 32H02 
