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26. CMB 2009 (vol 52 pp. 613)

Wulan, Hasi; Zhu, Kehe
 Lipschitz Type Characterizations for Bergman Spaces We obtain new characterizations for Bergman spaces with standard weights in terms of Lipschitz type conditions in the Euclidean, hyperbolic, and pseudo-hyperbolic metrics. As a consequence, we prove optimal embedding theorems when an analytic function on the unit disk is symmetrically lifted to the bidisk. Keywords:Bergman spaces, hyperbolic metric, Lipschitz conditionCategory:32A36

27. CMB 2009 (vol 52 pp. 285)

Parker, John R.; Platis, Ioannis D.
 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

28. CMB 2009 (vol 52 pp. 175)

Biswas, Indranil
 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, connectionCategories:32L05, 14F05

29. CMB 2009 (vol 52 pp. 154)

Ye, Yasheng; Ru, Min
 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

30. CMB 2009 (vol 52 pp. 84)

Gauthier, P. M.; Zeron, E. S.
 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 spaceCategories:32A10, 32D99, 32H99

31. CMB 2008 (vol 51 pp. 618)

Valmorin, V.
 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 functionsCategories:32A60, 45P05, 46F30

32. CMB 2008 (vol 51 pp. 481)

Bayart, Frédéric
 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, universalityCategories:32A35, 30D50, 47B38

33. CMB 2008 (vol 51 pp. 195)

Chen, Huaihui; Gauthier, Paul
 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 operatorsCategories:32A18, 30H05

34. CMB 2008 (vol 51 pp. 21)

Baracco, Luca
 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

35. CMB 2007 (vol 50 pp. 243)

Langlands, Robert P.
 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

36. CMB 2007 (vol 50 pp. 3)

Basener, Richard F.
 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

37. CMB 2006 (vol 49 pp. 628)

Zeron, E. S.
 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, homotopyCategories:32E30, 32Q55

38. CMB 2006 (vol 49 pp. 508)

Cho, Hong Rae
 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

39. CMB 2006 (vol 49 pp. 381)

Girela, Daniel; Peláez, José Ángel
 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 $01$). 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 spacesCategories:30D50, 30D55, 32A36

40. CMB 2006 (vol 49 pp. 237)

Gauthier, P. M.; Zeron, E. S.
 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-homotopicCategories:32E30, 32C18

41. CMB 2006 (vol 49 pp. 256)

Neelon, Tejinder
 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 seriesCategories:32A05, 26E05

42. CMB 2006 (vol 49 pp. 72)

Dwilewicz, Roman J.
 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 bundlesCategories:32F20, 14F05, 32C16

43. 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, $s$-metricCategories:32F45, 32Q45

44. CMB 2005 (vol 48 pp. 500)

Baracco, Luca
 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 extensionCategories:32D10, 32V25

45. CMB 2005 (vol 48 pp. 473)

Zeron, E. S.
 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 periodsCategories:32Q55, 14F45

46. CMB 2005 (vol 48 pp. 409)

Gauthier, P. M.; Xiao, J.
 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 functionsCategories:32A35, 30D50, 47B38

47. 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

48. 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

49. 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

50. 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
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