51. 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{ast117} agrees with
$$
(2\pi i)^{n} (1)^{n(n1)/2} \int_X \colon H^{2n}_{DR} (X,{\mathbb
C}) \to {\mathbb C}
$$
under the Dolbeault isomorphism.
Categories:14F10, 32A25, 14A15, 14F05, 18E30 

52. 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 

53. 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 CalabiEckmann manifold the following result of Kalyan Mukherjea:
Let $V_n = \mathbb{S}^{2n+1} \times \mathbb{S}^{2n+1}$ denote a
CalabiEckmann manifold. If $f,g \colon V_n \longrightarrow
\mathbb{P}^n$ are any two holomorphic maps, at least one of them being
nonconstant, 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 

54. 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 operator Categories:32A70, 47G10 

55. 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 $n3$. 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 structure Categories:14D20, 32C35 

56. CMB 2002 (vol 45 pp. 80)
57. 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 onedimensional 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 

58. CMB 2001 (vol 44 pp. 105)
59. 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
$N2$. Besides, they asked whether closed subsets of $\C^2$
homeomorphic to the real line (the simplest 1dimensional sets) are
removable for holomorphic functions. In this paper we propose a
positive answer to that question.
Keywords:holomorphic function, removable set Category:32D20 

60. CMB 2000 (vol 43 pp. 294)
61. CMB 2000 (vol 43 pp. 174)
62. 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 

63. 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 

64. 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 

65. CMB 1998 (vol 41 pp. 129)
66. 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 

67. CMB 1997 (vol 40 pp. 129)
68. 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 
