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1. CJM Online first

da Silva, Genival; Kerr, Matt; Pearlstein, Gregory
 Arithmetic of degenerating principal variations of Hodge structure: examples arising from mirror symmetry and middle convolution We collect evidence in support of a conjecture of Griffiths, Green and Kerr on the arithmetic of extension classes of limiting mixed Hodge structures arising from semistable degenerations over a number field. After briefly summarizing how a result of Iritani implies this conjecture for a collection of hypergeometric Calabi-Yau threefold examples studied by Doran and Morgan, the authors investigate a sequence of (non-hypergeometric) examples in dimensions $1\leq d\leq6$ arising from Katz's theory of the middle convolution. A crucial role is played by the Mumford-Tate group (which is $G_{2}$) of the family of 6-folds, and the theory of boundary components of Mumford-Tate domains. Keywords:variation of Hodge structure, limiting mixed Hodge structure, Calabi-Yau variety, middle convolution, Mumford-Tate groupCategories:14D07, 14M17, 17B45, 20G99, 32M10, 32G20

2. CJM Online first

Roth, Oliver
 Pontryagin's maximum principle for the Loewner equation in higher dimensions In this paper we develop a variational method for the Loewner equation in higher dimensions. As a result we obtain a version of Pontryagin's maximum principle from optimal control theory for the Loewner equation in several complex variables. Based on recent work of Arosio, Bracci and Wold, we then apply our version of the Pontryagin maximum principle to obtain first-order necessary conditions for the extremal mappings for a wide class of extremal problems over the set of normalized biholomorphic mappings on the unit ball in $\mathbb{C}^n$. Keywords:univalent function, Loewner's equationCategories:32H02, 30C55, 49K15

3. CJM 2014 (vol 66 pp. 961)

Baird, Thomas
 Moduli Spaces of Vector Bundles over a Real Curve: $\mathbb Z/2$-Betti Numbers Moduli spaces of real bundles over a real curve arise naturally as Lagrangian submanifolds of the moduli space of semi-stable bundles over a complex curve. In this paper, we adapt the methods of Atiyah-Bott's Yang-Mills over a Riemann Surface'' to compute $\mathbb Z/2$-Betti numbers of these spaces. Keywords:cohomology of moduli spaces, holomorphic vector bundlesCategories:32L05, 14P25

4. CJM 2013 (vol 66 pp. 1413)

Zhang, Xi; Zhang, Xiangwen
 Generalized KÃ¤hler--Einstein Metrics and Energy Functionals In this paper, we consider a generalized KÃ¤hler-Einstein equation on KÃ¤hler manifold $M$. Using the twisted $\mathcal K$-energy introduced by Song and Tian, we show that the existence of generalized KÃ¤hler-Einstein metrics with semi-positive twisting $(1, 1)$-form $\theta$ is also closely related to the properness of the twisted $\mathcal K$-energy functional. Under the condition that the twisting form $\theta$ is strictly positive at a point or $M$ admits no nontrivial Hamiltonian holomorphic vector field, we prove that the existence of generalized KÃ¤hler-Einstein metric implies a Moser-Trudinger type inequality. Keywords:complex Monge--AmpÃ¨re equation, energy functional, generalized KÃ¤hler--Einstein metric, Moser--Trudinger type inequalityCategories:53C55, 32W20

5. CJM 2012 (vol 65 pp. 808)

Grandjean, Vincent
 On Hessian Limit Directions along Gradient Trajectories Given a non-oscillating gradient trajectory $|\gamma|$ of a real analytic function $f$, we show that the limit $\nu$ of the secants at the limit point $\mathbf{0}$ of $|\gamma|$ along the trajectory $|\gamma|$ is an eigen-vector of the limit of the direction of the Hessian matrix $\operatorname{Hess} (f)$ at $\mathbf{0}$ along $|\gamma|$. The same holds true at infinity if the function is globally sub-analytic. We also deduce some interesting estimates along the trajectory. Away from the ends of the ambient space, this property is of metric nature and still holds in a general Riemannian analytic setting. Keywords:gradient trajectories, non-oscillation, limit of Hessian directions, limit of secants, trajectories at infinityCategories:34A26, 34C08, 32Bxx, 32Sxx

6. CJM 2012 (vol 66 pp. 197)

 On Hyperbolicity of Domains with Strictly Pseudoconvex Ends This article establishes a sufficient condition for Kobayashi hyperbolicity of unbounded domains in terms of curvature. Specifically, when $\Omega\subset{\mathbb C}^{n}$ corresponds to a sub-level set of a smooth, real-valued function $\Psi$, such that the form $\omega = {\bf i}\partial\bar{\partial}\Psi$ is KÃ¤hler and has bounded curvature outside a bounded subset, then this domain admits a hermitian metric of strictly negative holomorphic sectional curvature. Keywords:Kobayashi-hyperbolicity, KÃ¤hler metric, plurisubharmonic functionCategories:32Q45, 32Q35

7. CJM 2012 (vol 65 pp. 721)

Adamus, Janusz; Randriambololona, Serge; Shafikov, Rasul
 Tameness of Complex Dimension in a Real Analytic Set Given a real analytic set $X$ in a complex manifold and a positive integer $d$, denote by $\mathcal A^d$ the set of points $p$ in $X$ at which there exists a germ of a complex analytic set of dimension $d$ contained in $X$. It is proved that $\mathcal A^d$ is a closed semianalytic subset of $X$. Keywords:complex dimension, finite type, semianalytic set, tamenessCategories:32B10, 32B20, 32C07, 32C25, 32V15, 32V40, 14P15

8. CJM 2012 (vol 64 pp. 721)

Achab, Dehbia; Faraut, Jacques
 Analysis of the Brylinski-Kostant Model for Spherical Minimal Representations We revisit with another view point the construction by R. Brylinski and B. Kostant of minimal representations of simple Lie groups. We start from a pair $(V,Q)$, where $V$ is a complex vector space and $Q$ a homogeneous polynomial of degree 4 on $V$. The manifold $\Xi$ is an orbit of a covering of ${\rm Conf}(V,Q)$, the conformal group of the pair $(V,Q)$, in a finite dimensional representation space. By a generalized Kantor-Koecher-Tits construction we obtain a complex simple Lie algebra $\mathfrak g$, and furthermore a real form ${\mathfrak g}_{\mathbb R}$. The connected and simply connected Lie group $G_{\mathbb R}$ with ${\rm Lie}(G_{\mathbb R})={\mathfrak g}_{\mathbb R}$ acts unitarily on a Hilbert space of holomorphic functions defined on the manifold $\Xi$. Keywords:minimal representation, Kantor-Koecher-Tits construction, Jordan algebra, Bernstein identity, Meijer $G$-functionCategories:17C36, 22E46, 32M15, 33C80

9. CJM 2011 (vol 64 pp. 1329)

Izuchi, Kei Ji; Nguyen, Quang Dieu; Ohno, Shûichi
 Composition Operators Induced by Analytic Maps to the Polydisk We study properties of composition operators induced by symbols acting from the unit disk to the polydisk. This result will be involved in the investigation of weighted composition operators on the Hardy space on the unit disk and moreover be concerned with composition operators acting from the Bergman space to the Hardy space on the unit disk. Keywords:composition operators, Hardy spaces, polydiskCategories:47B33, 32A35, 30H10

10. CJM 2011 (vol 64 pp. 429)

Shafikov, Rasul; Verma, Kaushal
 Holomorphic Mappings between Domains in $\mathbb C^2$ An extension theorem for holomorphic mappings between two domains in $\mathbb C^2$ is proved under purely local hypotheses. Keywords:reflection principle, Segre varietiesCategories:32H40, 32H40

11. CJM 2011 (vol 63 pp. 755)

Chu, Kenneth C. K.
 On the Geometry of the Moduli Space of Real Binary Octics The moduli space of smooth real binary octics has five connected components. They parametrize the real binary octics whose defining equations have $0,\dots,4$ complex-conjugate pairs of roots respectively. We show that each of these five components has a real hyperbolic structure in the sense that each is isomorphic as a real-analytic manifold to the quotient of an open dense subset of $5$-dimensional real hyperbolic space $\mathbb{RH}^5$ by the action of an arithmetic subgroup of $\operatorname{Isom}(\mathbb{RH}^5)$. These subgroups are commensurable to discrete hyperbolic reflection groups, and the Vinberg diagrams of the latter are computed. Keywords:real binary octics, moduli space, complex hyperbolic geometry, Vinberg algorithmCategories:32G13, 32G20, 14D05, 14D20

12. CJM 2011 (vol 63 pp. 1038)

Cohen, D.; Denham, G.; Falk, M.; Varchenko, A.
 Critical Points and Resonance of Hyperplane Arrangements If $\Phi_\lambda$ is a master function corresponding to a hyperplane arrangement $\mathcal A$ and a collection of weights $\lambda$, we investigate the relationship between the critical set of $\Phi_\lambda$, the variety defined by the vanishing of the one-form $\omega_\lambda=\operatorname{d} \log \Phi_\lambda$, and the resonance of $\lambda$. For arrangements satisfying certain conditions, we show that if $\lambda$ is resonant in dimension $p$, then the critical set of $\Phi_\lambda$ has codimension at most $p$. These include all free arrangements and all rank $3$ arrangements. Keywords:hyperplane arrangement, master function, resonant weights, critical setCategories:32S22, 55N25, 52C35

13. CJM 2010 (vol 62 pp. 1276)

El Wassouli, Fouzia
 A Generalized Poisson Transform of an $L^{p}$-Function over the Shilov Boundary of the $n$-Dimensional Lie Ball Let $\mathcal{D}$ be the $n$-dimensional Lie ball and let $\mathbf{B}(S)$ be the space of hyperfunctions on the Shilov boundary $S$ of $\mathcal{D}$. The aim of this paper is to give a necessary and sufficient condition on the generalized Poisson transform $P_{l,\lambda}f$ of an element $f$ in the space $\mathbf{B}(S)$ for $f$ to be in $L^{p}(S)$, $1 > p > \infty.$ Namely, if $F$ is the Poisson transform of some $f\in \mathbf{B}(S)$ (i.e., $F=P_{l,\lambda}f$), then for any $l\in \mathbb{Z}$ and $\lambda\in \mathbb{C}$ such that $\mathcal{R}e[i\lambda] > \frac{n}{2}-1$, we show that $f\in L^{p}(S)$ if and only if $f$ satisfies the growth condition $$\|F\|_{\lambda,p}=\sup_{0\leq r < 1}(1-r^{2})^{\mathcal{R}e[i\lambda]-\frac{n}{2}+l}\Big[\int_{S}|F(ru)|^{p}du \Big]^{\frac{1}{p}} < +\infty.$$ Keywords:Lie ball, Shilov boundary, Fatou's theorem, hyperfuctions, parabolic subgroup, homogeneous line bundleCategories:32A45, 30E20, 33C67, 33C60, 33C55, 32A25, 33C75, 11K70

14. CJM 2010 (vol 62 pp. 889)

Xia, Jingbo
 Singular Integral Operators and Essential Commutativity on the Sphere Let ${\mathcal T}$ be the $C^\ast$-algebra generated by the Toeplitz operators $\{T_\varphi : \varphi \in L^\infty (S,d\sigma )\}$ on the Hardy space $H^2(S)$ of the unit sphere in $\mathbf{C}^n$. It is well known that ${\mathcal T}$ is contained in the essential commutant of $\{T_\varphi : \varphi \in \operatorname{VMO}\cap L^\infty (S,d\sigma )\}$. We show that the essential commutant of $\{T_\varphi : \varphi \in \operatorname{VMO}\cap L^\infty (S,d\sigma )\}$ is strictly larger than ${\mathcal T}$. Categories:32A55, 46L05, 47L80

15. CJM 2009 (vol 62 pp. 439)

Sundhäll, Marcus; Tchoundja, Edgar
 On Hankel Forms of Higher Weights: The Case of Hardy Spaces In this paper we study bilinear Hankel forms of higher weights on Hardy spaces in several dimensions. (The Schatten class Hankel forms of higher weights on weighted Bergman spaces have already been studied by Janson and Peetre for one dimension and by SundhÃ¤ll for several dimensions). We get a full characterization of Schatten class Hankel forms in terms of conditions for the symbols to be in certain Besov spaces. Also, the Hankel forms are bounded and compact if and only if the symbols satisfy certain Carleson measure criteria and vanishing Carleson measure criteria, respectively. Keywords:Hankel forms, Schattenâvon Neumann classes, Bergman spaces, Hardy spaces, Besov spaces, transvectant, unitary representations, MÃ¶bius groupCategories:32A25, 32A35, 32A37, 47B35

16. CJM 2009 (vol 62 pp. 218)

Xing, Yang
 The General Definition of the Complex Monge--AmpÃ¨re Operator on Compact KÃ¤hler Manifolds We introduce a wide subclass ${\mathcal F}(X,\omega)$ of quasi-plurisubharmonic functions in a compact KÃ¤hler manifold, on which the complex Monge-AmpÃ¨re operator is well defined and the convergence theorem is valid. We also prove that ${\mathcal F}(X,\omega)$ is a convex cone and includes all quasi-plurisubharmonic functions that are in the Cegrell class. Keywords:complex Monge--AmpÃ¨re operator, compact KÃ¤hler manifoldCategories:32W20, 32Q15

17. CJM 2009 (vol 62 pp. 3)

Anchouche, Boudjemâa
 On the Asymptotic Behavior of Complete KÃ¤hler Metrics of Positive Ricci Curvature Let $( X,g)$ be a complete noncompact KÃ¤hler manifold, of dimension $n\geq2,$ with positive Ricci curvature and of standard type (see the definition below). N. Mok proved that $X$ can be compactified, \emph{i.e.,} $X$ is biholomorphic to a quasi-projective variety$.$ The aim of this paper is to prove that the $L^{2}$ holomorphic sections of the line bundle $K_{X}^{-q}$ and the volume form of the metric $g$ have no essential singularities near the divisor at infinity. As a consequence we obtain a comparison between the volume forms of the KÃ¤hler metric $g$ and of the Fubini--Study metric induced on $X$. In the case of $\dim_{\mathbb{C} }X=2,$ we establish a relation between the number of components of the divisor $D$ and the dimension of the groups $H^{i}( \overline{X}, \Omega_{\overline{X}}^{1}( \log D) )$. Categories:53C55, 32A10

18. CJM 2009 (vol 61 pp. 1407)

Will, Pierre
 Traces, Cross-Ratios and 2-Generator Subgroups of $\SU(2,1)$ In this work, we investigate how to decompose a pair $(A,B)$ of loxodromic isometries of the complex hyperbolic plane $\mathbf H^{2}_{\mathbb C}$ under the form $A=I_1I_2$ and $B=I_3I_2$, where the $I_k$'s are involutions. The main result is a decomposability criterion, which is expressed in terms of traces of elements of the group $\langle A,B\rangle$. Categories:14L24, 22E40, 32M15, 51M10

19. 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

20. 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 $0 Categories:47B33, 32A70, 46E15 21. CJM 2008 (vol 60 pp. 1219) Baracco, Luca; Zampieri, Giuseppe  CR Extension from Manifolds of Higher Type This paper deals with the extension of CR functions from a manifold$M\subset \mathbb C^n$into directions produced by higher order commutators of holomorphic and antiholomorphic vector fields. It uses the theory of complex sectors'' attached to real submanifolds introduced in recent joint work of the authors with D. Zaitsev. In addition, it develops a new technique of approximation of sectors by smooth discs. Categories:32V25, 32V35, 32C16, 32F18 22. CJM 2008 (vol 60 pp. 721) Adamus, J.; Bierstone, E.; Milman, P. D.  Uniform Linear Bound in Chevalley's Lemma We obtain a uniform linear bound for the Chevalley function at a point in the source of an analytic mapping that is regular in the sense of Gabrielov. There is a version of Chevalley's lemma also along a fibre, or at a point of the image of a proper analytic mapping. We get a uniform linear bound for the Chevalley function of a closed Nash (or formally Nash) subanalytic set. Keywords:Chevalley function, regular mapping, Nash subanalytic setCategories:13J07, 32B20, 13J10, 32S10 23. CJM 2008 (vol 60 pp. 33) Braun, Rüdiger W.; Meise, Reinhold; Taylor, B. A.  Higher Order Tangents to Analytic Varieties along Curves. II Let~$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 $of algebraic varieties W along real analytic curves tending to infinity are derived by a reduction to the local case. Category:32C25 24. CJM 2007 (vol 59 pp. 1121) Alayont, Feryâl  Meromorphic Continuation of Spherical Cuspidal Data Eisenstein Series Meromorphic continuation of the Eisenstein series induced from spherical, cuspidal data on parabolic subgroups is achieved via reworking Bernstein's adaptation of Selberg's third proof of meromorphic continuation. Categories:11F72, 32N10, 32D15 25. CJM 2007 (vol 59 pp. 1098) Rodrigues, B.  Ruled Exceptional Surfaces and the Poles of Motivic Zeta Functions In this paper we study ruled surfaces which appear as an exceptional surface in a succession of blowing-ups. In particular we prove that the$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
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