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51. CMB 2004 (vol 47 pp. 133)

Wang, Wei
 Embeddability of Some Three-Dimensional Weakly Pseudoconvex ${\rm CR}$ Structures We prove that a class of perturbations of standard ${\rm CR}$ structure on the boundary of three-dimensional complex ellipsoid $E_{p,q}$ can be realized as hypersurfaces on $\mathbb{C}^2$, which generalizes the result of Burns and Epstein on the embeddability of some perturbations of standard ${\rm CR}$ structure on $S^3$. Keywords:deformations, embeddability, complex ellipsoidsCategories:32V30, 32G07, 32V35

52. CMB 2003 (vol 46 pp. 559)

Marco, Nicolas; Massaneda, Xavier
 On Density Conditions for Interpolation in the Ball In this paper we study interpolating sequences for two related spaces of holomorphic functions in the unit ball of $\C^n$, $n>1$. We first give density conditions for a sequence to be interpolating for the class $A^{-\infty}$ of holomorphic functions with polynomial growth. The sufficient condition is formally identical to the characterizing condition in dimension $1$, whereas the necessary one goes along the lines of the results given by Li and Taylor for some spaces of entire functions. In the second part of the paper we show that a density condition, which for $n=1$ coincides with the characterizing condition given by Seip, is sufficient for interpolation in the (weighted) Bergman space. Categories:32A36, 32A38, 30E05

53. CMB 2003 (vol 46 pp. 321)

Ballico, E.
 Discreteness For the Set of Complex Structures On a Real Variety Let $X$, $Y$ be reduced and irreducible compact complex spaces and $S$ the set of all isomorphism classes of reduced and irreducible compact complex spaces $W$ such that $X\times Y \cong X\times W$. Here we prove that $S$ is at most countable. We apply this result to show that for every reduced and irreducible compact complex space $X$ the set $S(X)$ of all complex reduced compact complex spaces $W$ with $X\times X^\sigma \cong W\times W^\sigma$ (where $A^\sigma$ denotes the complex conjugate of any variety $A$) is at most countable. Categories:32J18, 14J99, 14P99

54. CMB 2003 (vol 46 pp. 429)

Sastry, Pramathanath; Tong, Yue Lin L.
 The Grothendieck Trace and the de Rham Integral On a smooth $n$-dimensional complete variety $X$ over ${\mathbb C}$ we show that the trace map ${\tilde\theta}_X \colon\break H^n (X,\Omega_X^n) \to {\mathbb C}$ arising from Lipman's version of Grothendieck duality in \cite{ast-117} agrees with $$(2\pi i)^{-n} (-1)^{n(n-1)/2} \int_X \colon H^{2n}_{DR} (X,{\mathbb C}) \to {\mathbb C}$$ under the Dolbeault isomorphism. Categories:14F10, 32A25, 14A15, 14F05, 18E30

55. CMB 2003 (vol 46 pp. 291)

Sankaran, Parameswaran
 A Coincidence Theorem for Holomorphic Maps to $G/P$ The purpose of this note is to extend to an arbitrary generalized Hopf and Calabi-Eckmann manifold the following result of Kalyan Mukherjea: Let $V_n = \mathbb{S}^{2n+1} \times \mathbb{S}^{2n+1}$ denote a Calabi-Eckmann manifold. If $f,g \colon V_n \longrightarrow \mathbb{P}^n$ are any two holomorphic maps, at least one of them being non-constant, then there exists a coincidence: $f(x)=g(x)$ for some $x\in V_n$. Our proof involves a coincidence theorem for holomorphic maps to complex projective varieties of the form $G/P$ where $G$ is complex simple algebraic group and $P\subset G$ is a maximal parabolic subgroup, where one of the maps is dominant. Categories:32H02, 54M20

56. CMB 2003 (vol 46 pp. 113)

Lee, Jaesung; Rim, Kyung Soo
 Properties of the $\mathcal{M}$-Harmonic Conjugate Operator We define the $\mathcal{M}$-harmonic conjugate operator $K$ and prove that it is bounded on the nonisotropic Lipschitz space and on $\BMO$. Then we show $K$ maps Dini functions into the space of continuous functions on the unit sphere. We also prove the boundedness and compactness properties of $\mathcal{M}$-harmonic conjugate operator with $L^p$ symbol. Keywords:$\mathcal{M}$-harmonic conjugate operatorCategories:32A70, 47G10

57. CMB 2002 (vol 45 pp. 417)

Kamiyama, Yasuhiko; Tsukuda, Shuichi
 On Deformations of the Complex Structure on the Moduli Space of Spatial Polygons For an integer $n \geq 3$, let $M_n$ be the moduli space of spatial polygons with $n$ edges. We consider the case of odd $n$. Then $M_n$ is a Fano manifold of complex dimension $n-3$. Let $\Theta_{M_n}$ be the sheaf of germs of holomorphic sections of the tangent bundle $TM_n$. In this paper, we prove $H^q (M_n,\Theta_{M_n})=0$ for all $q \geq 0$ and all odd $n$. In particular, we see that the moduli space of deformations of the complex structure on $M_n$ consists of a point. Thus the complex structure on $M_n$ is locally rigid. Keywords:polygon space, complex structureCategories:14D20, 32C35

58. CMB 2002 (vol 45 pp. 80)

Gauthier, P. M.; Zeron, E. S.
 Approximation On Arcs and Dendrites Going to Infinity in $\C^n$ On a locally rectifiable arc going to infinity, each continuous function can be approximated by entire functions. Keywords:tangential approximation, CarlemanCategories:32E30, 32E25

59. CMB 2001 (vol 44 pp. 150)

Jakóbczak, Piotr
 Exceptional Sets of Slices for Functions From the Bergman Space in the Ball Let $B_N$ be the unit ball in $\mathbb{C}^N$ and let $f$ be a function holomorphic and $L^2$-integrable in $B_N$. Denote by $E(B_N,f)$ the set of all slices of the form $\Pi =L\cap B_N$, where $L$ is a complex one-dimensional subspace of $\mathbb{C}^N$, for which $f|_{\Pi}$ is not $L^2$-integrable (with respect to the Lebesgue measure on $L$). Call this set the exceptional set for $f$. We give a characterization of exceptional sets which are closed in the natural topology of slices. Categories:32A37, 32A22

60. CMB 2001 (vol 44 pp. 105)

Pilipović, Stevan
 Convolution Equation in $\mathcal{S}^{\prime\ast}$---Propagation of Singularities The singular spectrum of $u$ in a convolution equation $\mu * u = f$, where $\mu$ and $f$ are tempered ultradistributions of Beurling or Roumieau type is estimated by $$SS u \subset (\mathbf{R}^n \times \Char \mu) \cup SS f.$$ The same is done for $SS_{*}u$. Categories:32A40, 46F15, 58G07

61. CMB 2001 (vol 44 pp. 126)

Zeron, E. Santillan
 Each Copy of the Real Line in $\C^2$ is Removable Around 1995, Professors Lupacciolu, Chirka and Stout showed that a closed subset of $\C^N$ ($N\geq 2$) is removable for holomorphic functions, if its topological dimension is less than or equal to $N-2$. Besides, they asked whether closed subsets of $\C^2$ homeomorphic to the real line (the simplest 1-dimensional sets) are removable for holomorphic functions. In this paper we propose a positive answer to that question. Keywords:holomorphic function, removable setCategory:32D20

62. CMB 2000 (vol 43 pp. 294)

Bracci, Filippo
 Fixed Points of Commuting Holomorphic Maps Without Boundary Regularity We identify a class of domains of $\C^n$ in which any two commuting holomorphic self-maps have a common fixed point. Keywords:Holomorphic self-maps, commuting functions, fixed points, Wolff point, Julia's LemmaCategories:32A10, 32A40, 32H15, 32A30

63. 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

64. CMB 2000 (vol 43 pp. 47)

 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

65. 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

66. 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$, Hardy space, Bloch functionCategory:32A37 67. CMB 1998 (vol 41 pp. 129) Lee, Young Joo  Pluriharmonic symbols of commuting Toeplitz type operators on the weighted Bergman spaces A class of Toeplitz type operators acting on the weighted Bergman spaces of the unit ball in the$n$-dimensional complex space is considered and two pluriharmonic symbols of commuting Toeplitz type operators are completely characterized. Keywords:Pluriharmonic functions, Weighted Bergman spaces, Toeplitz type operators.Categories:47B38, 32A37 68. CMB 1997 (vol 40 pp. 356) Mazet, Pierre  Principe du maximum et lemme de Schwarz, a valeurs vectorielles Nous {\'e}tablissons un th{\'e}or{\e}me pour les fonctions holomorphes {\a} valeurs dans une partie convexe ferm{\'e}e. Ce th{\'e}or{\e}me pr{\'e}cise la position des coefficients de Taylor de telles fonctions et peut {\^e}tre consid{\'e}r{\'e} comme une g{\'e}n{\'e}ralisation des in{\'e}galit{\'e}s de Cauchy. Nous montrons alors comment ce th{\'e}or{\e}me permet de retrouver des versions connues du principe du maximum et d'obtenir de nouveaux r{\'e}sultats sur les applications holomorphes {\`a} valeurs vectorielles. Keywords:Principe du maximum, lemme de Schwarz, points extr{Ã©maux.Categories:30C80, 32A30, 46G20, 52A07 69. CMB 1997 (vol 40 pp. 129) Badea, Catalin  Sur les caractÃ¨res d'une algÃ¨bre de Banach A new proof for the Gleason-Kahane-\.Zelazko theorem concerning the characters of a Banach algebra is given. A theorem due to P\'olya and Saxer is used instead of the Hadamard factorization theorem. Categories:46H05, 32A15 70. CMB 1997 (vol 40 pp. 117) Vigué, Jean-Pierre  Un lemme de Schwarz pour les boules-unitÃ©s ouvertes Let$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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