Expand all Collapse all | Results 26 - 50 of 65 |
26. CMB 2009 (vol 52 pp. 285)
Global Geometrical Coordinates on Falbel's Cross-Ratio Variety Falbel has shown that four pairwise distinct points on the boundary
of a
complex hyperbolic $2$-space are completely determined, up to conjugation
in ${\rm PU}(2,1)$, by three complex cross-ratios satisfying two real
equations. We give global geometrical coordinates on the resulting
variety.
Categories:32G05, 32M05 |
27. CMB 2009 (vol 52 pp. 175)
Connections on a Parabolic Principal Bundle, II In \emph{Connections on a parabolic principal bundle over a curve, I}
we defined connections on a parabolic
principal bundle. While connections on usual principal bundles are
defined as splittings of the Atiyah exact sequence, it was noted in
the above article that the Atiyah exact sequence does not generalize to
the parabolic principal bundles.
Here we show that a twisted version
of the Atiyah exact sequence generalizes to the context of
parabolic principal bundles. For usual principal bundles, giving a
splitting of this twisted Atiyah exact sequence is equivalent
to giving a splitting of the Atiyah exact sequence. Connections on
a parabolic principal bundle can be defined using the
generalization of the twisted Atiyah exact sequence.
Keywords:Parabolic bundle, Atiyah exact sequence, connection Categories:32L05, 14F05 |
28. CMB 2009 (vol 52 pp. 154)
A Big Picard Theorem for Holomorphic Maps into Complex Projective Space We prove a big Picard type extension theorem for holomorphic maps
$f\from X-A \rightarrow M$, where $X$ is a complex manifold,
$A$ is an analytic subvariety of $X$, and $M$ is the complement of the
union of a set of hyperplanes in ${\Bbb P}^n$ but is not
necessarily hyperbolically imbedded in ${\Bbb P}^n$.
Category:32H30 |
29. CMB 2009 (vol 52 pp. 84)
Hartogs' Theorem on Separate Holomorphicity for Projective Spaces If a mapping of several complex variables into projective space is
holomorphic in each pair of variables, then it is globally
holomorphic.
Keywords:separately holomorphic, projective space Categories:32A10, 32D99, 32H99 |
30. CMB 2008 (vol 51 pp. 481)
Universal Inner Functions on the Ball It is shown that given any sequence of automorphisms $(\phi_k)_k$ of the
unit ball $\bn$ of $\cn$ such that $\|\phi_k(0)\|$ tends to $1$,
there exists an inner function
$I$ such that the family of ``non-Euclidean translates"
$(I\circ\phi_k)_k$ is locally uniformly dense in the unit ball of
$H^\infty(\bn)$.
Keywords:inner functions, automorphisms of the ball, universality Categories:32A35, 30D50, 47B38 |
31. CMB 2008 (vol 51 pp. 618)
Vanishing Theorems in Colombeau Algebras of Generalized Functions Using a canonical linear embedding of the algebra
${\mathcal G}^{\infty}(\Omega)$ of Colombeau generalized functions in the space of
$\overline{\C}$-valued $\C$-linear maps on the space
${\mathcal D}(\Omega)$ of smooth functions with compact support, we give vanishing
conditions for functions and linear integral operators of class
${\mathcal G}^\infty$. These results are then applied to the zeros of holomorphic
generalized functions in dimension greater than one.
Keywords:Colombeau generalized functions, linear integral operators, generalized holomorphic functions Categories:32A60, 45P05, 46F30 |
32. CMB 2008 (vol 51 pp. 195)
Boundedness from Below of Composition Operators on $\alpha$-Bloch Spaces We give a necessary and sufficient condition for a composition
operator on an $\alpha$-Bloch space with $\alpha\ge 1$ to be bounded below.
This extends a known result for the Bloch space due to P. Ghatage,
J. Yan, D. Zheng, and H. Chen.
Keywords:Bloch functions, composition operators Categories:32A18, 30H05 |
33. CMB 2008 (vol 51 pp. 21)
A Remark on Extensions of CR Functions from Hyperplanes In the characterization of the range of the Radon transform, one
encounters the problem of the holomorphic extension of functions
defined on $\R^2\setminus\Delta_\R$ (where $\Delta_\R$ is the diagonal
in $\R^2$) and which extend as ``separately holomorphic" functions of
their two arguments. In particular, these functions extend in fact to $\C^2\setminus
\Delta_\C$ where $\Delta_\C$ is the complexification of
$\Delta_\R$. We take this theorem from the integral geometry and put
it in the more natural context of the CR geometry where it accepts an
easier proof and a more general statement. In this new setting it
becomes a variant of the celebrated ``edge of the wedge" theorem of
Ajrapetyan and Henkin.
Categories:32D10, 32V25 |
34. CMB 2007 (vol 50 pp. 243)
Un nouveau point de repÃ¨re dans la thÃ©orie des formes automorphes Dans le papier Beyond Endoscopy une id\'ee pour entamer la
fonctorialit\'e en utilisant la formule des traces a \'et\'e
introduite. Maints probl\`emes, l'existence d'une limite convenable
de la formule des traces, est eqquiss\'ee dans cette note
informelle mais seulement pour $GL(2)$ et les corps des fonctions
rationelles sur un corps fini et en ne pas resolvant
bon nombre de questions.
Categories:32N10, 14xx |
35. CMB 2007 (vol 50 pp. 3)
Higher Dimensional Spaces of Functions on the Spectrum of a Uniform Algebra In this paper we introduce a nested family of spaces of continuous functions defined
on the spectrum of a uniform algebra. The smallest space in the family is the
uniform algebra itself. In the ``finite dimensional'' case, from some point on the
spaces will be the space of all continuous complex-valued functions on the
spectrum. These spaces are defined in terms of solutions to the nonlinear
Cauchy--Riemann equations as introduced by the author in 1976, so they are not
generally linear spaces of functions. However, these spaces do shed light on the
higher dimensional properties of a uniform algebra. In particular, these spaces are
directly related to the generalized Shilov boundary of the uniform algebra (as
defined by the author and, independently, by Sibony in the early 1970s).
Categories:32A99, 46J10 |
36. CMB 2006 (vol 49 pp. 628)
Approximation and the Topology of Rationally Convex Sets Considering a mapping $g$ holomorphic on a neighbourhood of a rationally
convex set $K\subset\cc^n$, and range into the complex projective space
$\cc\pp^m$, the main objective of this paper is to show that we can
uniformly approximate $g$ on $K$ by rational mappings defined from
$\cc^n$ into $\cc\pp^m$. We only need to ask that the second \v{C}ech
cohomology group $\check{H}^2(K,\zz)$ vanishes.
Keywords:Rationally convex, cohomology, homotopy Categories:32E30, 32Q55 |
37. CMB 2006 (vol 49 pp. 508)
Growth Spaces and Growth Norm Estimates for $\Bar\partial$ on Convex Domains of Finite Type We consider the growth norm of a measurable function $f$ defined by
$$\|f\|_{-\sigma}=\ess\{\delta_D(z)^\sigma|f(z)|:z\in D\},$$
where $\delta_D(z)$ denote the distance from $z$ to $\partial D$.
We prove some optimal growth norm estimates for $\bar\partial$
on convex domains of finite type.
Categories:32W05, 32A26, 32A36 |
38. CMB 2006 (vol 49 pp. 381)
On the Membership in Bergman Spaces of the Derivative of a Blaschke Product With Zeros in a Stolz Domain |
On the Membership in Bergman Spaces of the Derivative of a Blaschke Product With Zeros in a Stolz Domain It is known that the derivative of
a Blaschke product whose zero sequence lies in a Stolz angle
belongs to all the Bergman spaces $A^p$ with $0
1$). As a consequence, we prove that there exists a Blaschke product $B$ with zeros on a radius such that $B'\notin A^{3/2}$. Keywords:Blaschke products, Hardy spaces, Bergman spaces Categories:30D50, 30D55, 32A36 |
39. CMB 2006 (vol 49 pp. 237)
Approximation by Rational Mappings, via Homotopy Theory Continuous mappings defined from compact subsets $K$ of complex
Euclidean space $\cc^n$ into complex projective space $\pp^m$ are
approximated by rational mappings. The fundamental tool employed
is homotopy theory.
Keywords:Rational approximation, homotopy type, null-homotopic Categories:32E30, 32C18 |
40. CMB 2006 (vol 49 pp. 256)
A Bernstein--Walsh Type Inequality and Applications A Bernstein--Walsh type inequality for $C^{\infty }$ functions of several
variables is derived, which then is applied to obtain analogs and
generalizations of the following classical theorems: (1) Bochnak--Siciak
theorem: a $C^{\infty }$\ function on $\mathbb{R}^{n}$ that is real
analytic on every line is real analytic; (2) Zorn--Lelong theorem: if a
double power series $F(x,y)$\ converges on a set of lines of positive
capacity then $F(x,y)$\ is convergent; (3) Abhyankar--Moh--Sathaye theorem:
the transfinite diameter of the convergence set of a divergent series is
zero.
Keywords:Bernstein-Walsh inequality, convergence sets, analytic functions, ultradifferentiable functions, formal power series Categories:32A05, 26E05 |
41. CMB 2006 (vol 49 pp. 72)
Additive Riemann--Hilbert Problem in Line Bundles Over $\mathbb{CP}^1$ In this note we consider $\overline\partial$-problem in
line bundles over complex projective space $\mathbb{CP}^1$
and prove that the
equation can be solved for $(0,1)$ forms with compact support. As a
consequence, any Cauchy-Riemann function on a compact real hypersurface in
such line bundles is a jump of two holomorphic functions defined on the
sides of the hypersurface. In particular, the results can be applied to
$\mathbb{CP}^2$ since by removing a point from it we get a line bundle over
$\mathbb{CP}^1$.
Keywords:$\overline\partial$-problem, cohomology groups, line bundles Categories:32F20, 14F05, 32C16 |
42. CMB 2005 (vol 48 pp. 601)
On the Regularity of the $s$-Differential Metric We show that the injective Kobayashi--Royden differential metric, as
defined by Hahn, is upper semicontinous.
Keywords:Invariant metric, Kobayashi--Royden metric, Hahn metric, $s$-metric Categories:32F45, 32Q45 |
43. CMB 2005 (vol 48 pp. 500)
Extension of Holomorphic Functions From One Side of a Hypersurface We give a new proof of former results by G. Zampieri and the
author on extension of holomorphic
functions from one side $\Omega$ of a real hypersurface
$M$ of $\mathbb{C}^n$ in the presence of an
analytic disc tangent to $M$, attached to $\bar\Omega$
but not to $M$. Our method enables
us to weaken the regularity assumptions both
for the hypersurface and the disc.
Keywords:analytic discs, Poisson integral, holomorphic extension Categories:32D10, 32V25 |
44. CMB 2005 (vol 48 pp. 473)
Logarithms and the Topology of the Complement of a Hypersurface This paper is devoted to analysing the relation between the
logarithm of a non-constant holomorphic polynomial $Q(z)$ and
the topology of the complement of the hypersurface defined by
$Q(z)=0$.
Keywords:Logarithm, homology groups and periods Categories:32Q55, 14F45 |
45. CMB 2005 (vol 48 pp. 409)
The Existence of Universal Inner Functions on the Unit Ball of $\mathbb{C}^n$ It is shown that there exists an inner function
$I$ defined on the unit ball ${\bf B}^n$ of ${\mathbb C}^n$
such that each function holomorphic on ${\bf B}^n$ and
bounded by $1$ can be approximated by
``non-Euclidean translates" of $I$.
Keywords:universal inner functions Categories:32A35, 30D50, 47B38 |
46. CMB 2004 (vol 47 pp. 133)
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 ellipsoids Categories:32V30, 32G07, 32V35 |
47. CMB 2003 (vol 46 pp. 559)
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 |
48. CMB 2003 (vol 46 pp. 321)
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 |
49. CMB 2003 (vol 46 pp. 429)
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 |
50. CMB 2003 (vol 46 pp. 291)
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 |