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26. CJM 2009 (vol 61 pp. 566)

Graham, Ian; Hamada, Hidetaka; Kohr, Gabriela; Pfaltzgraff, John A.
 Convex Subordination Chains in Several Complex Variables In this paper we study the notion of a convex subordination chain in several complex variables. We obtain certain necessary and sufficient conditions for a mapping to be a convex subordination chain, and we give various examples of convex subordination chains on the Euclidean unit ball in $\mathbb{C}^n$. We also obtain a sufficient condition for injectivity of $f(z/\|z\|,\|z\|)$ on $B^n\setminus\{0\}$, where $f(z,t)$ is a convex subordination chain over $(0,1)$. Keywords:biholomorphic mapping, convex mapping, convex subordination chain, Loewner chain, subordinationCategories:32H02, 30C45

27. CJM 2009 (vol 61 pp. 50)

Chen, Huaihui; Gauthier, Paul
 Composition operators on $\mu$-Bloch spaces Given a positive continuous function $\mu$ on the interval $0 Categories:47B33, 32A70, 46E15 28. CJM 2008 (vol 60 pp. 1219) Baracco, Luca; Zampieri, Giuseppe  CR Extension from Manifolds of Higher Type This paper deals with the extension of CR functions from a manifold$M\subset \mathbb C^n$into directions produced by higher order commutators of holomorphic and antiholomorphic vector fields. It uses the theory of complex sectors'' attached to real submanifolds introduced in recent joint work of the authors with D. Zaitsev. In addition, it develops a new technique of approximation of sectors by smooth discs. Categories:32V25, 32V35, 32C16, 32F18 29. CJM 2008 (vol 60 pp. 721) Adamus, J.; Bierstone, E.; Milman, P. D.  Uniform Linear Bound in Chevalley's Lemma We obtain a uniform linear bound for the Chevalley function at a point in the source of an analytic mapping that is regular in the sense of Gabrielov. There is a version of Chevalley's lemma also along a fibre, or at a point of the image of a proper analytic mapping. We get a uniform linear bound for the Chevalley function of a closed Nash (or formally Nash) subanalytic set. Keywords:Chevalley function, regular mapping, Nash subanalytic setCategories:13J07, 32B20, 13J10, 32S10 30. CJM 2008 (vol 60 pp. 33) Braun, Rüdiger W.; Meise, Reinhold; Taylor, B. A.  Higher Order Tangents to Analytic Varieties along Curves. II Let~$V$be an analytic variety in some open set in~$\C^n$. For a real analytic curve~$\gamma$with$ \gamma(0) = 0 $and$ d \ge 1 $define$ V_t = t^{-d}(V - \gamma(t)) $. It was shown in a previous paper that the currents of integration over~$V_t$converge to a limit current whose support$ T_{\gamma,d} V $is an algebraic variety as~$t$tends to zero. Here, it is shown that the canonical defining function of the limit current is the suitably normalized limit of the canonical defining functions of the~$V_t$. As a corollary, it is shown that$ T_{\gamma,d} V $is either inhomogeneous or coincides with$ T_{\gamma,\delta} V $for all~$\delta$in some neighborhood of~$d$. As another application it is shown that for surfaces only a finite number of curves lead to limit varieties that are interesting for the investigation of Phragm\'en--Lindel\"of conditions. Corresponding results for limit varieties$ T_{\sigma,\delta} W $of algebraic varieties W along real analytic curves tending to infinity are derived by a reduction to the local case. Category:32C25 31. CJM 2007 (vol 59 pp. 1121) Alayont, Feryâl  Meromorphic Continuation of Spherical Cuspidal Data Eisenstein Series Meromorphic continuation of the Eisenstein series induced from spherical, cuspidal data on parabolic subgroups is achieved via reworking Bernstein's adaptation of Selberg's third proof of meromorphic continuation. Categories:11F72, 32N10, 32D15 32. CJM 2007 (vol 59 pp. 1098) Rodrigues, B.  Ruled Exceptional Surfaces and the Poles of Motivic Zeta Functions In this paper we study ruled surfaces which appear as an exceptional surface in a succession of blowing-ups. In particular we prove that the$e$-invariant of such a ruled exceptional surface$E$is strictly positive whenever its intersection with the other exceptional surfaces does not contain a fiber (of$E$). This fact immediately enables us to resolve an open problem concerning an intersection configuration on such a ruled exceptional surface consisting of three nonintersecting sections. In the second part of the paper we apply the non-vanishing of$e$to the study of the poles of the well-known topological, Hodge and motivic zeta functions. Categories:14E15, 14J26, 14B05, 14J17, 32S45 33. CJM 2007 (vol 59 pp. 1069) Reydy, Carine  Quotients jacobiens : une approche algÃ©brique Le diagramme d'Eisenbud et Neumann d'un germe est un arbre qui repr\'esente ce germe et permet d'en calculer les invariants. On donne une d\'emonstration alg\'ebrique d'un r\'esultat caract\'erisant l'ensemble des quotients jacobiens d'un germe d'application$(f,g)$\a partir du diagramme d'Eisenbud et Neumann de$fg$. Keywords:SingularitÃ©, jacobien, quotient jacobien, polygone de NewtonCategories:14B05, 32S05, 32S50 34. CJM 2007 (vol 59 pp. 3) Biller, Harald  Holomorphic Generation of Continuous Inverse Algebras We study complex commutative Banach algebras (and, more generally, continuous inverse algebras) in which the holomorphic functions of a fixed$n$-tuple of elements are dense. In particular, we characterize the compact subsets of~$\C^n$which appear as joint spectra of such$n$-tuples. The characterization is compared with several established notions of holomorphic convexity by means of approximation conditions. Keywords:holomorphic functional calculus, commutative continuous inverse algebra, holomorphic convexity, Stein compacta, meromorphic convexity, holomorphic approximationCategories:46H30, 32A38, 32E30, 41A20, 46J15 35. CJM 2006 (vol 58 pp. 262) Biswas, Indranil  Connections on a Parabolic Principal Bundle Over a Curve The aim here is to define connections on a parabolic principal bundle. Some applications are given. Keywords:parabolic bundle, holomorphic connection, unitary connectionCategories:53C07, 32L05, 14F05 36. CJM 2005 (vol 57 pp. 506) Gross, Leonard; Grothaus, Martin  Reverse Hypercontractivity for Subharmonic Functions Contractivity and hypercontractivity properties of semigroups are now well understood when the generator,$A$, is a Dirichlet form operator. It has been shown that in some holomorphic function spaces the semigroup operators,$e^{-tA}$, can be bounded {\it below} from$L^p$to$L^q$when$p,q$and$t$are suitably related. We will show that such lower boundedness occurs also in spaces of subharmonic functions. Keywords:Reverse hypercontractivity, subharmonicCategories:58J35, 47D03, 47D07, 32Q99, 60J35 37. CJM 2005 (vol 57 pp. 3) Alberich-Carramiñana, Maria; Roé, Joaquim  Enriques Diagrams and Adjacency of Planar Curve Singularities We study adjacency of equisingularity types of planar complex curve singularities in terms of their Enriques diagrams. The goal is, given two equisingularity types, to determine whether one of them is adjacent to the other. For linear adjacency a complete answer is obtained, whereas for arbitrary (analytic) adjacency a necessary condition and a sufficient condition are proved. We also obtain new examples of exceptional deformations, {\em i.e.,} singular curves of type$\mathcal{D}'$that can be deformed to a curve of type$\mathcal{D}$without$\mathcal{D}'$being adjacent to$\mathcal{D}$. 38. CJM 2003 (vol 55 pp. 64) Braun, Rüdiger W.; Meise, Reinhold; Taylor, B. A.  Higher Order Tangents to Analytic Varieties along Curves Let$V$be an analytic variety in some open set in$\mathbb{C}^n$which contains the origin and which is purely$k$-dimensional. For a curve$\gamma$in$\mathbb{C}^n$, defined by a convergent Puiseux series and satisfying$\gamma(0) = 0$, and$d \ge 1$, define$V_t := t^{-d} \bigl( V-\gamma(t) \bigr)$. Then the currents defined by$V_t$converge to a limit current$T_{\gamma,d} [V]$as$t$tends to zero.$T_{\gamma,d} [V]$is either zero or its support is an algebraic variety of pure dimension$k$in$\mathbb{C}^n$. Properties of such limit currents and examples are presented. These results will be applied in a forthcoming paper to derive necessary conditions for varieties satisfying the local Phragm\'en-Lindel\"of condition that was used by H\"ormander to characterize the constant coefficient partial differential operators which act surjectively on the space of all real analytic functions on$\mathbb{R}^n$. Category:32C25 39. CJM 2002 (vol 54 pp. 1254) Isaev, A. V.; Kruzhilin, N. G.  Effective Actions of the Unitary Group on Complex Manifolds We classify all connected$n$-dimensional complex manifolds admitting effective actions of the unitary group$U_n$by biholomorphic transformations. One consequence of this classification is a characterization of$\CC^n$by its automorphism group. Keywords:complex manifolds, group actionsCategories:32Q57, 32M17 40. CJM 2002 (vol 54 pp. 493) Braden, Tom  Perverse Sheaves on Grassmannians We compute the category of perverse sheaves on Hermitian symmetric spaces in types~A and D, constructible with respect to the Schubert stratification. The calculation is microlocal, and uses the action of the Borel group to study the geometry of the conormal variety$\Lambda$. Keywords:perverse sheaves, microlocal geometryCategories:32S60, 32C38, 35A27 41. CJM 2002 (vol 54 pp. 324) Graham, Ian; Hamada, Hidetaka; Kohr, Gabriela  Parametric Representation of Univalent Mappings in Several Complex Variables Let$B$be the unit ball of$\bb{C}^n$with respect to an arbitrary norm. We prove that the analog of the Carath\'eodory set, {\it i.e.} the set of normalized holomorphic mappings from$B$into$\bb{C}^n$of positive real part'', is compact. This leads to improvements in the existence theorems for the Loewner differential equation in several complex variables. We investigate a subset of the normalized biholomorphic mappings of$B$which arises in the study of the Loewner equation, namely the set$S^0(B)$of mappings which have parametric representation. For the case of the unit polydisc these mappings were studied by Poreda, and on the Euclidean unit ball they were studied by Kohr. As in Kohr's work, we consider subsets of$S^0(B)$obtained by placing restrictions on the mapping from the Carath\'eodory set which occurs in the Loewner equation. We obtain growth and covering theorems for these subsets of$S^0(B)$as well as coefficient estimates, and consider various examples. Also we shall see that in higher dimensions there exist mappings in$S(B)$which can be imbedded in Loewner chains, but which do not have parametric representation. Categories:32H02, 30C45 42. CJM 2002 (vol 54 pp. 55) Ban, Chunsheng; McEwan, Lee J.; Némethi, András  On the Milnor Fiber of a Quasi-ordinary Surface Singularity We verify a generalization of (3.3) from \cite{Le} proving that the homotopy type of the Milnor fiber of a reduced hypersurface singularity depends only on the embedded topological type of the singularity. In particular, using \cite{Za,Li1,Oh1,Gau} for irreducible quasi-ordinary germs, it depends only on the normalized distinguished pairs of the singularity. The main result of the paper provides an explicit formula for the Euler-characteristic of the Milnor fiber in the surface case. Categories:14B05, 14E15, 32S55 43. CJM 2001 (vol 53 pp. 834) Veys, Willem  Zeta Functions and Kontsevich Invariants' on Singular Varieties Let$X$be a nonsingular algebraic variety in characteristic zero. To an effective divisor on$X$Kontsevich has associated a certain motivic integral, living in a completion of the Grothendieck ring of algebraic varieties. He used this invariant to show that birational (smooth, projective) Calabi-Yau varieties have the same Hodge numbers. Then Denef and Loeser introduced the invariant {\it motivic (Igusa) zeta function}, associated to a regular function on$X$, which specializes to both the classical$p$-adic Igusa zeta function and the topological zeta function, and also to Kontsevich's invariant. This paper treats a generalization to singular varieties. Batyrev already considered such a Kontsevich invariant' for log terminal varieties (on the level of Hodge polynomials of varieties instead of in the Grothendieck ring), and previously we introduced a motivic zeta function on normal surface germs. Here on any$\bbQ$-Gorenstein variety$X$we associate a motivic zeta function and a Kontsevich invariant' to effective$\bbQ$-Cartier divisors on$X$whose support contains the singular locus of~$X$. Keywords:singularity invariant, topological zeta function, motivic zeta functionCategories:14B05, 14E15, 32S50, 32S45 44. CJM 2001 (vol 53 pp. 73) Fukui, Toshizumi; Paunescu, Laurentiu  Stratification Theory from the Weighted Point of View In this paper, we investigate stratification theory in terms of the defining equations of strata and maps (without tube systems), offering a concrete approach to show that some given family is topologically trivial. In this approach, we consider a weighted version of$(w)$-regularity condition and Kuo's ratio test condition. Categories:32B99, 14P25, 32Cxx, 58A35 45. CJM 2000 (vol 52 pp. 1149) Ban, Chunsheng; McEwan, Lee J.  Canonical Resolution of a Quasi-ordinary Surface Singularity We describe the embedded resolution of an irreducible quasi-ordinary surface singularity$(V,p)$which results from applying the canonical resolution of Bierstone-Milman to$(V,p)$. We show that this process depends solely on the characteristic pairs of$(V,p)$, as predicted by Lipman. We describe the process explicitly enough that a resolution graph for$f$could in principle be obtained by computer using only the characteristic pairs. Keywords:canonical resolution, quasi-ordinary singularityCategories:14B05, 14J17, 32S05, 32S25 46. CJM 2000 (vol 52 pp. 982) Lárusson, Finnur  Holomorphic Functions of Slow Growth on Nested Covering Spaces of Compact Manifolds Let$Y$be an infinite covering space of a projective manifold$M$in$\P^N$of dimension$n\geq 2$. Let$C$be the intersection with$M$of at most$n-1$generic hypersurfaces of degree$d$in$\mathbb{P}^N$. The preimage$X$of$C$in$Y$is a connected submanifold. Let$\phi$be the smoothed distance from a fixed point in$Y$in a metric pulled up from$M$. Let$\O_\phi(X)$be the Hilbert space of holomorphic functions$f$on$X$such that$f^2 e^{-\phi}$is integrable on$X$, and define$\O_\phi(Y)$similarly. Our main result is that (under more general hypotheses than described here) the restriction$\O_\phi(Y) \to \O_\phi(X)$is an isomorphism for$d$large enough. This yields new examples of Riemann surfaces and domains of holomorphy in$\C^n$with corona. We consider the important special case when$Y$is the unit ball$\B$in$\C^n$, and show that for$d$large enough, every bounded holomorphic function on$X$extends to a unique function in the intersection of all the nontrivial weighted Bergman spaces on$\B$. Finally, assuming that the covering group is arithmetic, we establish three dichotomies concerning the extension of bounded holomorphic and harmonic functions from$X$to$\B$. Categories:32A10, 14E20, 30F99, 32M15 47. CJM 2000 (vol 52 pp. 1085) Xing, Yang  Complex Monge-AmpÃ¨re Measures of Plurisubharmonic Functions with Bounded Values Near the Boundary We give a characterization of bounded plurisubharmonic functions by using their complex Monge-Amp\ere measures. This implies a both necessary and sufficient condition for a positive measure to be complex Monge-Amp\ere measure of some bounded plurisubharmonic function. Categories:32F07, 32F05 48. CJM 2000 (vol 52 pp. 348) González Pérez, P. D.  SingularitÃ©s quasi-ordinaires toriques et polyÃ¨dre de Newton du discriminant Nous \'etudions les polyn\^omes$F \in \C \{S_\tau\} [Y] $\a coefficients dans l'anneau de germes de fonctions holomorphes au point sp\'ecial d'une vari\'et\'e torique affine. Nous g\'en\'eralisons \a ce cas la param\'etrisation classique des singularit\'es quasi-ordinaires. Cela fait intervenir d'une part une g\'en\'eralization de l'algorithme de Newton-Puiseux, et d'autre part une relation entre le poly\edre de Newton du discriminant de$F$par rapport \a$Y$et celui de$F$au moyen du polytope-fibre de Billera et Sturmfels~\cite{Sturmfels}. Cela nous permet enfin de calculer, sous des hypoth\eses de non d\'eg\'en\'erescence, les sommets du poly\edre de Newton du discriminant a partir de celui de$F$, et les coefficients correspondants \a partir des coefficients des exposants de$F$qui sont dans les ar\^etes de son poly\edre de Newton. Categories:14M25, 32S25 49. CJM 2000 (vol 52 pp. 3) Aizenberg, Lev; Vidras, Alekos  On Small Complete Sets of Functions Using Local Residues and the Duality Principle a multidimensional variation of the completeness theorems by T.~Carleman and A.~F.~Leontiev is proven for the space of holomorphic functions defined on a suitable open strip$T_{\alpha}\subset {\bf C}^2$. The completeness theorem is a direct consequence of the Cauchy Residue Theorem in a torus. With suitable modifications the same result holds in${\bf C}^n$. Categories:32A10, 42C30 50. CJM 1999 (vol 51 pp. 915) Balogh, Zoltán M.; Leuenberger, Christoph  Quasiconformal Contactomorphisms and Polynomial Hulls with Convex Fibers Consider the polynomial hull of a smoothly varying family of strictly convex smooth domains fibered over the unit circle. It is well-known that the boundary of the hull is foliated by graphs of analytic discs. We prove that this foliation is smooth, and we show that it induces a complex flow of contactomorphisms. These mappings are quasiconformal in the sense of Kor\'anyi and Reimann. A similar bound on their quasiconformal distortion holds as in the one-dimensional case of holomorphic motions. The special case when the fibers are rotations of a fixed domain in$\C^2\$ is studied in details. Categories:32E20, 30C65
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