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51. CMB 2000 (vol 43 pp. 174)

Gantz, Christian; Steer, Brian
 Stable Parabolic Bundles over Elliptic Surfaces and over Riemann Surfaces We show that the use of orbifold bundles enables some questions to be reduced to the case of flat bundles. The identification of moduli spaces of certain parabolic bundles over elliptic surfaces is achieved using this method. Categories:14J27, 32L07, 14H60, 14D20

52. 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 micro-support of a $G$-equivariant sheaf on $X$ is contained in the conormal variety of the $G$-action. Categories:32B20, 22E15

53. 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 $A-D-E$ singularities. We give two new characterizations of simple isolated line singularities. Categories:32S25, 14B05

54. 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$, Hardy space, Bloch functionCategory:32A37 55. CMB 1998 (vol 41 pp. 129) Lee, Young Joo  Pluriharmonic symbols of commuting Toeplitz type operators on the weighted Bergman spaces A class of Toeplitz type operators acting on the weighted Bergman spaces of the unit ball in the$n$-dimensional complex space is considered and two pluriharmonic symbols of commuting Toeplitz type operators are completely characterized. Keywords:Pluriharmonic functions, Weighted Bergman spaces, Toeplitz type operators.Categories:47B38, 32A37 56. CMB 1997 (vol 40 pp. 356) Mazet, Pierre  Principe du maximum et lemme de Schwarz, a valeurs vectorielles Nous {\'e}tablissons un th{\'e}or{\e}me pour les fonctions holomorphes {\a} valeurs dans une partie convexe ferm{\'e}e. Ce th{\'e}or{\e}me pr{\'e}cise la position des coefficients de Taylor de telles fonctions et peut {\^e}tre consid{\'e}r{\'e} comme une g{\'e}n{\'e}ralisation des in{\'e}galit{\'e}s de Cauchy. Nous montrons alors comment ce th{\'e}or{\e}me permet de retrouver des versions connues du principe du maximum et d'obtenir de nouveaux r{\'e}sultats sur les applications holomorphes {\`a} valeurs vectorielles. Keywords:Principe du maximum, lemme de Schwarz, points extr{Ã©maux.Categories:30C80, 32A30, 46G20, 52A07 57. CMB 1997 (vol 40 pp. 129) Badea, Catalin  Sur les caractÃ¨res d'une algÃ¨bre de Banach A new proof for the Gleason-Kahane-\.Zelazko theorem concerning the characters of a Banach algebra is given. A theorem due to P\'olya and Saxer is used instead of the Hadamard factorization theorem. Categories:46H05, 32A15 58. CMB 1997 (vol 40 pp. 117) Vigué, Jean-Pierre  Un lemme de Schwarz pour les boules-unitÃ©s ouvertes Let$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
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