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1. CMB Online first

Gupta, Purvi
A real-analytic nonpolynomially convex isotropic torus with no attached discs
We show by means of an example in $\mathbb C^3$ that Gromov's theorem on the presence of attached holomorphic discs for compact Lagrangian manifolds is not true in the subcritical real-analytic case, even in the absence of an obvious obstruction, i.e, polynomial convexity.

Keywords:polynomial hull, isotropic submanifold, holomorphic disc
Categories:32V40, 32E20, 53D12

2. CMB Online first

Sebbar, Abdellah; Al-Shbeil, Isra
Elliptic Zeta functions and equivariant functions
In this paper we establish a close connection between three notions attached to a modular subgroup. Namely the set of weight two meromorphic modular forms, the set of equivariant functions on the upper half-plane commuting with the action of the modular subgroup and the set of elliptic zeta functions generalizing the Weierstrass zeta functions. In particular, we show that the equivariant functions can be parameterized by modular objects as well as by elliptic objects.

Keywords:modular form, equivariant function, elliptic zeta function
Categories:11F12, 35Q15, 32L10

3. CMB Online first

Bayart, Frédéric; Gauthier, Paul M
Functions universal for all translation operators in several complex variables
We prove the existence of a (in fact many) holomorphic function $f$ in $\mathbb{C}^d$ such that, for any $a\neq 0$, its translations $f(\cdot+na)$ are dense in $H(\mathbb{C}^d)$.

Keywords:hypercyclic operator, translation operator
Categories:47A16, 32E20

4. CMB Online first

Ding, Fan; Geiges, Hansjörg; Zhang, Guangjian
On subcritically Stein fillable 5-manifolds
We make some elementary observations concerning subcritically Stein fillable contact structures on $5$-manifolds. Specifically, we determine the diffeomorphism type of such contact manifolds in the case the fundamental group is finite cyclic, and we show that on the $5$-sphere the standard contact structure is the unique subcritically fillable one. More generally, it is shown that subcritically fillable contact structures on simply connected $5$-manifolds are determined by their underlying almost contact structure. Along the way, we discuss the homotopy classification of almost contact structures.

Keywords:subcritically Stein fillable, 5-manifold, almost contact structure, thickening
Categories:53D35, 32Q28, 57M20, 57Q10, 57R17

5. CMB Online first

Gilligan, Bruce
Levi's problem for pseudoconvex homogeneous manifolds
Suppose $G$ is a connected complex Lie group and $H$ is a closed complex subgroup. Then there exists a closed complex subgroup $J$ of $G$ containing $H$ such that the fibration $\pi:G/H \to G/J$ is the holomorphic reduction of $G/H$, i.e., $G/J$ is holomorphically separable and ${\mathcal O}(G/H) \cong \pi^*{\mathcal O}(G/J)$. In this paper we prove that if $G/H$ is pseudoconvex, i.e., if $G/H$ admits a continuous plurisubharmonic exhaustion function, then $G/J$ is Stein and $J/H$ has no non--constant holomorphic functions.

Keywords:complex homogeneous manifold, plurisubharmonic exhaustion function, holomorphic reduction, Stein manifold, Remmert reduction, Hirschowitz annihilator
Categories:32M10, 32U10, 32A10, 32Q28

6. CMB Online first

Li, Bao Qin
An Equivalent Form of Picard's Theorem and Beyond
This paper gives an equivalent form of Picard's theorem via entire solutions of the functional equation $f^2+g^2=1$, and then its improvements and applications to certain nonlinear (ordinary and partial) differential equations.

Keywords:entire function, Picard's Theorem, functional equation, partial differential equation
Categories:30D20, 32A15, 35F20

7. CMB 2017 (vol 60 pp. 381)

Rousseau, C.
The Bifurcation Diagram of Cubic Polynomial Vector Fields on $\mathbb{C}\mathbb{P}^1$
In this paper we give the bifurcation diagram of the family of cubic vector fields $\dot z=z^3+ \epsilon_1z+\epsilon_0$ for $z\in \mathbb{C}\mathbb{P}^1$, depending on the values of $\epsilon_1,\epsilon_0\in\mathbb{C}$. The bifurcation diagram is in $\mathbb{R}^4$, but its conic structure allows describing it for parameter values in $\mathbb{S}^3$. There are two open simply connected regions of structurally stable vector fields separated by surfaces corresponding to bifurcations of homoclinic connections between two separatrices of the pole at infinity. These branch from the codimension 2 curve of double singular points. We also explain the bifurcation of homoclinic connection in terms of the description of Douady and Sentenac of polynomial vector fields.

Keywords:complex polynomial vector field, bifurcation diagram, Douady-Sentenac invariant
Categories:34M45, 32G34

8. CMB Online first

Miranda-Neto, Cleto Brasileiro
A module-theoretic characterization of algebraic hypersurfaces
In this note we prove the following surprising characterization: if $X\subset {\mathbb A}^n$ is an (embedded, non-empty, proper) algebraic variety defined over a field $k$ of characteristic zero, then $X$ is a hypersurface if and only if the module $T_{{\mathcal O}_{{\mathbb A}^n}/k}(X)$ of logarithmic vector fields of $X$ is a reflexive ${\mathcal O}_{{\mathbb A}^n}$-module. As a consequence of this result, we derive that if $T_{{\mathcal O}_{{\mathbb A}^n}/k}(X)$ is a free ${\mathcal O}_{{\mathbb A}^n}$-module, which is shown to be equivalent to the freeness of the $t$th exterior power of $T_{{\mathcal O}_{{\mathbb A}^n}/k}(X)$ for some (in fact, any) $t\leq n$, then necessarily $X$ is a Saito free divisor.

Keywords:hypersurface, logarithmic vector field, logarithmic derivation, free divisor
Categories:14J70, 13N15, 32S22, 13C05, 13C10, 14N20, , , , , 14C20, 32M25

9. CMB 2016 (vol 59 pp. 449)

Abdallah, Nancy
On Hodge Theory of Singular Plane Curves
The dimensions of the graded quotients of the cohomology of a plane curve complement $U=\mathbb P^2 \setminus C$ with respect to the Hodge filtration are described in terms of simple geometrical invariants. The case of curves with ordinary singularities is discussed in detail. We also give a precise numerical estimate for the difference between the Hodge filtration and the pole order filtration on $H^2(U,\mathbb C)$.

Keywords:plane curves, Hodge and pole order filtrations
Categories:32S35, 32S22, 14H50

10. CMB 2016 (vol 59 pp. 279)

Dimca, Alexandru
The Poincaré-Deligne Polynomial of Milnor Fibers of Triple Point Line Arrangements is Combinatorially Determined
Using a recent result by S. Papadima and A. Suciu, we show that the equivariant Poincaré-Deligne polynomial of the Milnor fiber of a projective line arrangement having only double and triple points is combinatorially determined.

Keywords:line arrangement, Milnor fiber, monodromy, mixed Hodge structures
Categories:32S22, 32S35, 32S25, 32S55

11. CMB 2016 (vol 59 pp. 346)

Krantz, Steven
On a Theorem of Bers, with Applications to the Study of Automorphism Groups of Domains
We study and generalize a classical theorem of L. Bers that classifies domains up to biholomorphic equivalence in terms of the algebras of holomorphic functions on those domains. Then we develop applications of these results to the study of domains with noncompact automorphism group.

Keywords:Bers's theorem, algebras of holomorphic functions, noncompact automorphism group, biholomorphic equivalence
Categories:32A38, 30H50, 32A10, 32M99

12. CMB 2015 (vol 59 pp. 182)

Naylor, Geoff; Rolfsen, Dale
Generalized Torsion in Knot Groups
In a group, a nonidentity element is called a generalized torsion element if some product of its conjugates equals the identity. We show that for many classical knots one can find generalized torsion in the fundamental group of its complement, commonly called the knot group. It follows that such a group is not bi-orderable. Examples include all torus knots, the (hyperbolic) knot $5_2$ and algebraic knots in the sense of Milnor.

Keywords:knot group, generalized torsion, ordered group
Categories:57M27, 32S55, 29F60

13. CMB 2015 (vol 58 pp. 281)

Kalus, Matthias
On the Relation of Real and Complex Lie Supergroups
A complex Lie supergroup can be described as a real Lie supergroup with integrable almost complex structure. The necessary and sufficient conditions on an almost complex structure on a real Lie supergroup for defining a complex Lie supergroup are deduced. The classification of real Lie supergroups with such almost complex structures yields a new approach to the known classification of complex Lie supergroups by complex Harish-Chandra superpairs. A universal complexification of a real Lie supergroup is constructed.

Keywords:Lie supergroup, almost complex structure, Harish-Chandra pair, universal complexification
Categories:32C11, 58A50

14. CMB 2015 (vol 58 pp. 381)

Tang, Xiaomin; Liu, Taishun
The Schwarz Lemma at the Boundary of the Egg Domain $B_{p_1, p_2}$ in $\mathbb{C}^n$
Let $B_{p_1, p_2}=\{z\in\mathbb{C}^n: |z_1|^{p_1}+|z_2|^{p_2}+\cdots+|z_n|^{p_2}\lt 1\}$ be an egg domain in $\mathbb{C}^n$. In this paper, we first characterize the Kobayashi metric on $B_{p_1, p_2}\,(p_1\geq 1, p_2\geq 1)$, and then establish a new type of the classical boundary Schwarz lemma at $z_0\in\partial{B_{p_1, p_2}}$ for holomorphic self-mappings of $B_{p_1, p_2}(p_1\geq 1, p_2\gt 1)$, where $z_0=(e^{i\theta}, 0, \dots, 0)'$ and $\theta\in \mathbb{R}$.

Keywords:holomorphic mapping, Schwarz lemma, Kobayashi metric, egg domain
Categories:32H02, 30C80, 32A30

15. CMB 2014 (vol 57 pp. 697)

Bailet, Pauline
On the Monodromy of Milnor Fibers of Hyperplane Arrangements
We describe a general setting where the monodromy action on the first cohomology group of the Milnor fiber of a hyperplane arrangement is the identity.

Keywords:hyperplane arrangements, Milnor fiber, monodromy, local systems
Categories:32S22, 32S55, 32S25, 32S40

16. CMB 2014 (vol 57 pp. 658)

Thang, Nguyen Tat
Admissibility of Local Systems for some Classes of Line Arrangements
Let $\mathcal{A}$ be a line arrangement in the complex projective plane $\mathbb{P}^2$ and let $M$ be its complement. A rank one local system $\mathcal{L}$ on $M$ is admissible if roughly speaking the cohomology groups $H^m(M,\mathcal{L})$ can be computed directly from the cohomology algebra $H^{*}(M,\mathbb{C})$. In this work, we give a sufficient condition for the admissibility of all rank one local systems on $M$. As a result, we obtain some properties of the characteristic variety $\mathcal{V}_1(M)$ and the Resonance variety $\mathcal{R}_1(M)$.

Keywords:admissible local system, line arrangement, characteristic variety, multinet, resonance variety
Categories:14F99, 32S22, 52C35, 05A18, 05C40, 14H50

17. CMB 2014 (vol 57 pp. 673)

Ahmadi, S. Ruhallah; Gilligan, Bruce
Complexifying Lie Group Actions on Homogeneous Manifolds of Non-compact Dimension Two
If $X$ is a connected complex manifold with $d_X = 2$ that admits a (connected) Lie group $G$ acting transitively as a group of holomorphic transformations, then the action extends to an action of the complexification $\widehat{G}$ of $G$ on $X$ except when either the unit disk in the complex plane or a strictly pseudoconcave homogeneous complex manifold is the base or fiber of some homogeneous fibration of $X$.

Keywords:homogeneous complex manifold, non-compact dimension two, complexification

18. CMB 2013 (vol 57 pp. 870)

Parlier, Hugo
A Short Note on Short Pants
It is a theorem of Bers that any closed hyperbolic surface admits a pants decomposition consisting of curves of bounded length where the bound only depends on the topology of the surface. The question of the quantification of the optimal constants has been well studied and the best upper bounds to date are linear in genus, a theorem of Buser and Seppälä. The goal of this note is to give a short proof of a linear upper bound which slightly improve the best known bound.

Keywords:hyperbolic surfaces, geodesics, pants decompositions
Categories:30F10, 32G15, 53C22

19. CMB 2013 (vol 57 pp. 794)

Fang, Zhong-Shan; Zhou, Ze-Hua
New Characterizations of the Weighted Composition Operators Between Bloch Type Spaces in the Polydisk
We give some new characterizations for compactness of weighted composition operators $uC_\varphi$ acting on Bloch-type spaces in terms of the power of the components of $\varphi,$ where $\varphi$ is a holomorphic self-map of the polydisk $\mathbb{D}^n,$ thus generalizing the results obtained by Hyvärinen and Lindström in 2012.

Keywords:weighted composition operator, compactness, Bloch type spaces, polydisk, several complex variables
Categories:47B38, 47B33, 32A37, 45P05, 47G10

20. CMB 2012 (vol 57 pp. 12)

Aribi, Amine; Dragomir, Sorin; El Soufi, Ahmad
On the Continuity of the Eigenvalues of a Sublaplacian
We study the behavior of the eigenvalues of a sublaplacian $\Delta_b$ on a compact strictly pseudoconvex CR manifold $M$, as functions on the set ${\mathcal P}_+$ of positively oriented contact forms on $M$ by endowing ${\mathcal P}_+$ with a natural metric topology.

Keywords:CR manifold, contact form, sublaplacian, Fefferman metric
Categories:32V20, 53C56

21. CMB 2011 (vol 56 pp. 593)

Liu, Congwen; Zhou, Lifang
On the $p$-norm of an Integral Operator in the Half Plane
We give a partial answer to a conjecture of Dostanić on the determination of the norm of a class of integral operators induced by the weighted Bergman projection in the upper half plane.

Keywords:Bergman projection, integral operator, $L^p$-norm, the upper half plane
Categories:47B38, 47G10, 32A36

22. CMB 2011 (vol 56 pp. 31)

Ayuso, Fortuny P.
Derivations and Valuation Rings
A complete characterization of valuation rings closed for a holomorphic derivation is given, following an idea of Seidenberg, in dimension $2$.

Keywords:singular holomorphic foliation, derivation, valuation, valuation ring
Categories:32S65, 13F30, 13A18

23. CMB 2011 (vol 56 pp. 44)

Biswas, Indranil; Dey, Arijit
Polystable Parabolic Principal $G$-Bundles and Hermitian-Einstein Connections
We show that there is a bijective correspondence between the polystable parabolic principal $G$-bundles and solutions of the Hermitian-Einstein equation.

Keywords:ramified principal bundle, parabolic principal bundle, Hitchin-Kobayashi correspondence, polystability
Categories:32L04, 53C07

24. CMB 2011 (vol 55 pp. 108)

Guler, Dincer
On Segre Forms of Positive Vector Bundles
The goal of this note is to prove that the signed Segre forms of Griffiths' positive vector bundles are positive.

Categories:53C55, 32L05

25. CMB 2011 (vol 55 pp. 329)

Kamiya, Shigeyasu; Parker, John R.; Thompson, James M.
Non-Discrete Complex Hyperbolic Triangle Groups of Type $(n,n, \infty;k)$
A complex hyperbolic triangle group is a group generated by three involutions fixing complex lines in complex hyperbolic space. Our purpose in this paper is to improve a previous result and to discuss discreteness of complex hyperbolic triangle groups of type $(n,n,\infty;k)$.

Keywords:complex hyperbolic triangle group
Categories:51M10, 32M15, 53C55, 53C35
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