1. CJM 2016 (vol 69 pp. 241)
 Adamus, Janusz; Seyedinejad, Hadi

Finite Determinacy and Stability of Flatness of Analytic Mappings
It is proved that flatness of an analytic mapping germ from a
complete intersection is determined by its sufficiently high
jet. As a consequence, one obtains finite determinacy of complete
intersections. It is also shown that flatness and openness are
stable under deformations.
Keywords:finite determinacy, stability, flatness, openness, complete intersection Categories:58K40, 58K25, 32S05, 58K20, 32S30, 32B99, 32C05, 13B40 

2. CJM 2016 (vol 68 pp. 1257)
3. CJM 2016 (vol 69 pp. 220)
 Zheng, Tao

The ChernRicci Flow on OeljeklausToma Manifolds
We study the ChernRicci flow, an evolution equation of Hermitian
metrics, on a family of OeljeklausToma (OT) manifolds which
are nonKÃ¤hler compact complex manifolds with negative Kodaira
dimension. We prove that, after an initial conformal change,
the flow converges, in the
GromovHausdorff sense, to a torus with a flat Riemannian metric
determined by the OTmanifolds themselves.
Keywords:ChernRicci flow, OeljeklausToma manifold, Calabitype estimate, GromovHausdorff convergence Categories:53C44, 53C55, 32W20, 32J18, 32M17 

4. CJM 2016 (vol 68 pp. 625)
 Ingram, Patrick

Rigidity and Height Bounds for Certain Postcritically Finite Endomorphisms of $\mathbb P^N$
The morphism $f:\mathbb{P}^N\to\mathbb{P}^N$ is called postcritically finite
(PCF) if the forward image of the critical locus, under iteration
of $f$, has algebraic support. In the case $N=1$, a result of
Thurston implies that there are no algebraic families of PCF
morphisms, other than a wellunderstood exceptional class known
as the flexible LattÃ¨s maps. A related arithmetic result
states that the set of PCF morphisms corresponds to a set of
bounded height in the moduli space of univariate rational functions.
We prove corresponding results for a certain subclass of the
regular polynomial endomorphisms of $\mathbb{P}^N$, for any $N$.
Keywords:postcritically finite, arithmetic dynamics, heights Categories:37P15, 32H50, 37P30 

5. CJM 2016 (vol 68 pp. 280)
 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
CalabiYau threefold examples studied by Doran and Morgan,
the authors investigate a sequence of (nonhypergeometric) examples
in dimensions $1\leq d\leq6$ arising from Katz's theory of the
middle
convolution.
A crucial role is played by the MumfordTate
group (which is $G_{2}$) of the family of 6folds, and the theory
of boundary components of MumfordTate domains.
Keywords:variation of Hodge structure, limiting mixed Hodge structure, CalabiYau variety, middle convolution, MumfordTate group Categories:14D07, 14M17, 17B45, 20G99, 32M10, 32G20 

6. CJM 2015 (vol 69 pp. 408)
 Klep, Igor; Špenko, Špela

Free Function Theory Through Matrix Invariants
This paper concerns free function theory. Free maps are free
analogs of analytic functions in several complex variables,
and are defined in terms of freely noncommuting variables.
A function of $g$ noncommuting variables is a function on $g$tuples
of square matrices of all sizes that respects direct sums and
simultaneous conjugation.
Examples of such maps include noncommutative polynomials, noncommutative
rational functions and convergent noncommutative power series.
In sharp contrast to the existing literature in free analysis, this article
investigates free maps \emph{with involution} 
free analogs of real analytic functions.
To
get a grip on these,
techniques and tools from invariant theory are developed and
applied to free analysis. Here is a sample of the results obtained.
A characterization of polynomial free maps via properties of
their finitedimensional slices is presented and then used to
establish power series expansions for analytic free maps about
scalar and nonscalar points; the latter are series of generalized
polynomials for which an invarianttheoretic characterization
is given.
Furthermore,
an inverse and implicit function theorem for free maps with
involution is obtained.
Finally, with a selection of carefully chosen examples
it is shown that
free maps with involution
do not exhibit strong rigidity properties
enjoyed by their involutionfree
counterparts.
Keywords:free algebra, free analysis, invariant theory, polynomial identities, trace identities, concomitants, analytic maps, inverse function theorem, generalized polynomials Categories:16R30, 32A05, 46L52, 15A24, 47A56, 15A24, 46G20 

7. CJM 2015 (vol 69 pp. 130)
 Levin, Aaron; Wang, Julie TzuYueh

On NonArchimedean Curves Omitting Few Components and their Arithmetic Analogues
Let $\mathbf{k}$ be an algebraically closed field complete with respect
to a nonArchimedean absolute value of arbitrary characteristic.
Let $D_1,\dots, D_n$ be effective nef divisors intersecting
transversally in an $n$dimensional nonsingular projective variety
$X$.
We study the degeneracy of nonArchimedean analytic maps from
$\mathbf{k}$ into $X\setminus \cup_{i=1}^nD_i$ under various geometric
conditions. When $X$ is a rational ruled surface and $D_1$ and
$D_2$ are ample, we obtain a necessary and sufficient condition
such that
there is no nonArchimedean analytic map from $\mathbf{k}$ into $X\setminus
D_1 \cup D_2$.
Using the dictionary between nonArchimedean Nevanlinna theory
and Diophantine approximation that originated in
earlier work with T. T. H. An, %
we also study arithmetic analogues of these problems, establishing
results on integral points on these varieties over $\mathbb{Z}$
or the ring of integers of an imaginary quadratic field.
Keywords:nonArchimedean Picard theorem, nonArchimedean analytic curves, integral points Categories:11J97, 32P05, 32H25 

8. CJM 2014 (vol 67 pp. 942)
 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 firstorder 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 equation Categories:32H02, 30C55, 49K15 

9. 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 semistable
bundles over a complex curve. In this paper, we adapt the methods
of AtiyahBott's ``YangMills over a Riemann Surface'' to compute
$\mathbb Z/2$Betti numbers of these spaces.
Keywords:cohomology of moduli spaces, holomorphic vector bundles Categories:32L05, 14P25 

10. CJM 2013 (vol 66 pp. 1413)
 Zhang, Xi; Zhang, Xiangwen

Generalized KÃ¤hlerEinstein Metrics and Energy Functionals
In this paper, we consider a generalized
KÃ¤hlerEinstein 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¤hlerEinstein metrics with
semipositive 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¤hlerEinstein metric implies a MoserTrudinger type inequality.
Keywords:complex MongeAmpÃ¨re equation, energy functional, generalized KÃ¤hlerEinstein metric, MoserTrudinger type inequality Categories:53C55, 32W20 

11. CJM 2012 (vol 65 pp. 808)
 Grandjean, Vincent

On Hessian Limit Directions along Gradient Trajectories
Given a nonoscillating 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 eigenvector 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 subanalytic. 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, nonoscillation, limit of Hessian directions, limit of secants, trajectories at infinity Categories:34A26, 34C08, 32Bxx, 32Sxx 

12. CJM 2012 (vol 66 pp. 197)
 Harris, Adam; Kolář, Martin

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
sublevel set of a smooth, realvalued 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:Kobayashihyperbolicity, KÃ¤hler metric, plurisubharmonic function Categories:32Q45, 32Q35 

13. 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, tameness Categories:32B10, 32B20, 32C07, 32C25, 32V15, 32V40, 14P15 

14. CJM 2012 (vol 64 pp. 721)
 Achab, Dehbia; Faraut, Jacques

Analysis of the BrylinskiKostant 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 KantorKoecherTits 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, KantorKoecherTits construction, Jordan algebra, Bernstein identity, Meijer $G$function Categories:17C36, 22E46, 32M15, 33C80 

15. CJM 2011 (vol 64 pp. 1329)
16. CJM 2011 (vol 64 pp. 429)
17. 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 oneform $\omega_\lambda=\operatorname{d} \log \Phi_\lambda$, 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 set Categories:32S22, 55N25, 52C35 

18. 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 $0,\dots,4$ complexconjugate 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
realanalytic 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 algorithm Categories:32G13, 32G20, 14D05, 14D20 

19. 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
$$
\F\_{\lambda,p}=\sup_{0\leq r
< 1}(1r^{2})^{\mathcal{R}e[i\lambda]\frac{n}{2}+l}\Big[\int_{S}F(ru)^{p}du
\Big]^{\frac{1}{p}} < +\infty.
$$
Keywords:Lie ball, Shilov boundary, Fatou's theorem, hyperfuctions, parabolic subgroup, homogeneous line bundle Categories:32A45, 30E20, 33C67, 33C60, 33C55, 32A25, 33C75, 11K70 

20. 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 

21. CJM 2009 (vol 62 pp. 218)
 Xing, Yang

The General Definition of the Complex MongeAmpÃ¨re Operator on Compact KÃ¤hler Manifolds
We introduce a wide subclass ${\mathcal F}(X,\omega)$ of
quasiplurisubharmonic functions in a compact KÃ¤hler manifold, on
which the complex MongeAmpÃ¨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 quasiplurisubharmonic functions
that are in the Cegrell class.
Keywords:complex MongeAmpÃ¨re operator, compact KÃ¤hler manifold Categories:32W20, 32Q15 

22. 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, Schattenâvon Neumann classes, Bergman spaces, Hardy spaces, Besov spaces, transvectant, unitary representations, MÃ¶bius group Categories:32A25, 32A35, 32A37, 47B35 

23. 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 quasiprojective
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 FubiniStudy
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 

24. CJM 2009 (vol 61 pp. 1407)
 Will, Pierre

Traces, CrossRatios and 2Generator 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 

25. 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, subordination Categories:32H02, 30C45 
