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1. CMB Online first

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 filtrationsCategories:32S35, 32S22, 14H50

2. CMB 2016 (vol 59 pp. 279)

Dimca, Alexandru
 The PoincarÃ©-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 PoincarÃ©-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 structuresCategories:32S22, 32S35, 32S25, 32S55

3. 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 equivalenceCategories:32A38, 30H50, 32A10, 32M99

4. 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 groupCategories:57M27, 32S55, 29F60

5. 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 complexificationCategories:32C11, 58A50

6. 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 domainCategories:32H02, 30C80, 32A30

7. 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 systemsCategories:32S22, 32S55, 32S25, 32S40

8. 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 varietyCategories:14F99, 32S22, 52C35, 05A18, 05C40, 14H50

9. 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, complexificationCategory:32M10

10. 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 SeppÃ¤lÃ¤. 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 decompositionsCategories:30F10, 32G15, 53C22

11. 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 HyvÃ¤rinen and LindstrÃ¶m in 2012. Keywords:weighted composition operator, compactness, Bloch type spaces, polydisk, several complex variablesCategories:47B38, 47B33, 32A37, 45P05, 47G10

12. 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 metricCategories:32V20, 53C56

13. 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 DostaniÄ 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 planeCategories:47B38, 47G10, 32A36

14. 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 ringCategories:32S65, 13F30, 13A18

15. 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, polystabilityCategories:32L04, 53C07

16. 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

17. 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 groupCategories:51M10, 32M15, 53C55, 53C35

18. CMB 2011 (vol 55 pp. 249)

Chang, Der-Chen; Li, Bao Qin
 Description of Entire Solutions of Eiconal Type Equations The paper describes entire solutions to the eiconal type non-linear partial differential equations, which include the eiconal equations $(X_1(u))^2+(X_2(u))^2=1$ as special cases, where $X_1=p_1{\partial}/{\partial z_1}+p_2{\partial}/{\partial z_2}$, $X_2=p_3{\partial}/{\partial z_1}+p_4{\partial}/{\partial z_2}$ are linearly independent operators with $p_j$ being arbitrary polynomials in $\mathbf{C}^2$. Keywords:entire solution, eiconal equation, polynomial, transcendental functionCategories:32A15, 35F20

19. CMB 2011 (vol 55 pp. 242)

Cegrell, Urban
 Convergence in Capacity In this note we study the convergence of sequences of Monge-AmpÃ¨re measures $\{(dd^cu_s)^n\}$, where $\{u_s\}$ is a given sequence of plurisubharmonic functions, converging in capacity. Keywords:complex Monge-AmpÃ¨re operator, convergence in capacity, plurisubharmonic functionCategories:32U20, 31C15

20. CMB 2011 (vol 55 pp. 441)

Zorboska, Nina
 Univalently Induced, Closed Range, Composition Operators on the Bloch-type Spaces While there is a large variety of univalently induced closed range composition operators on the Bloch space, we show that the only univalently induced, closed range, composition operators on the Bloch-type spaces $B^{\alpha}$ with $\alpha \ne 1$ are the ones induced by a disc automorphism. Keywords:composition operators, Bloch-type spaces, closed range, univalentCategories:47B35, 32A18

21. CMB 2011 (vol 55 pp. 146)

Li, Songxiao; Wulan, Hasi; Zhu, Kehe
 A Characterization of Bergman Spaces on the Unit Ball of ${\mathbb C}^n$. II It has been shown that a holomorphic function $f$ in the unit ball $\mathbb{B}_n$ of ${\mathbb C}_n$ belongs to the weighted Bergman space $A^p_\alpha$, $p>n+1+\alpha$, if and only if the function $|f(z)-f(w)|/|1-\langle z,w\rangle|$ is in $L^p(\mathbb{B}_n\times\mathbb{B}_n,dv_\beta \times dv_\beta)$, where $\beta=(p+\alpha-n-1)/2$ and $dv_\beta(z)= (1-|z|^2)^\beta\,dv(z)$. In this paper we consider the range $0n+1+\alpha$ is particularly interesting. Keywords:Bergman spaces, unit ball, volume measureCategory:32A36

22. CMB 2011 (vol 54 pp. 230)

Clouâtre, Raphaël
 Universal Power Series in $\mathbb{C}^N$ We establish the existence of power series in $\mathbb{C}^N$ with the property that the subsequences of the sequence of partial sums uniformly approach any holomorphic function on any well chosen compact subset outside the set of convergence of the series. We also show that, in a certain sense, most series enjoy this property. Categories:32A05, 32E30

23. CMB 2011 (vol 54 pp. 338)

Nakazi, Takahiko
 SzegÃ¶'s Theorem and Uniform Algebras We study SzegÃ¶'s theorem for a uniform algebra. In particular, we do it for the disc algebra or the bidisc algebra. Keywords:SzegÃ¶'s theorem, uniform algebras, disc algebra, weighted Bergman spaceCategories:32A35, 46J15, 60G25

24. CMB 2010 (vol 54 pp. 370)

Stout, Edgar Lee
 Manifold-Valued Holomorphic Approximation This note considers the problem of approximating continuous maps from sets in complex spaces into complex manifolds by holomorphic maps. Category:32E20

25. CMB 2010 (vol 54 pp. 56)

Dinh, Thi Anh Thu
 Characteristic Varieties for a Class of Line Arrangements Let $\mathcal{A}$ be a line arrangement in the complex projective plane $\mathbb{P}^2$, having the points of multiplicity $\geq 3$ situated on two lines in $\mathcal{A}$, say $H_0$ and $H_{\infty}$. Then we show that the non-local irreducible components of the first resonance variety $\mathcal{R}_1(\mathcal{A})$ are 2-dimensional and correspond to parallelograms $\mathcal{P}$ in $\mathbb{C}^2=\mathbb{P}^2 \setminus H_{\infty}$ whose sides are in $\mathcal{A}$ and for which $H_0$ is a diagonal. Keywords:local system, line arrangement, characteristic variety, resonance varietyCategories:14C21, 14F99, 32S22, 14E05, 14H50
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