On the fundamental group of the complement of a complex hyperplane arrangement

2018 ◽  
Author(s):  
Luis Paris
Keyword(s):  
Fundamental Group ◽  
Hyperplane Arrangement ◽  
Complex Hyperplane

Poincaré series of Milnor algebras and free arrangements

2010 ◽  
Vol 47 (4) ◽  
pp. 513-521
Author(s):  
Shahid Ahmad
Keyword(s):  
Hyperplane Arrangement ◽  
Betti Numbers ◽  
Poincaré Series ◽  
Poincare Series ◽  
Complex Hyperplane

We show that for a free complex hyperplane arrangement \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{A}$$ \end{document}: f = 0, the Poincaré series of the graded Milnor algebra M(f) and the Betti numbers of the arrangement complement M(\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{A}$$ \end{document}) determine each other. Examples show that this is false if we drop the freeness assumption.

scholarly journals Tate properties, polynomial-count varieties, and monodromy of hyperplane arrangements

2012 ◽  
Vol 206 ◽  
pp. 75-97 ◽  
Author(s):  
Alexandru Dimca
Keyword(s):  
Hyperplane Arrangement ◽  
Hyperplane Arrangements ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Monodromy Operator ◽  
Low Dimensions ◽  
Milnor Fiber ◽  
Necessary And Sufficient ◽  
Complex Hyperplane

AbstractThe order of the Milnor fiber monodromy operator of a central hyperplane arrangement is shown to be combinatorially determined. In particular, a necessary and sufficient condition for the triviality of this monodromy operator is given.It is known that the complement of a complex hyperplane arrangement is cohomologically Tate and, if the arrangement is defined over ℚ, has polynomial count. We show that these properties hold for the corresponding Milnor fibers if the monodromy is trivial.We construct a hyperplane arrangement defined over ℚ, whose Milnor fiber has a nontrivial monodromy operator, is cohomologically Tate, and has no polynomial count. Such examples are shown not to exist in low dimensions.

Cubature rules from Hall–Littlewood polynomials

2020 ◽  
Author(s):  
J F van Diejen ◽  
E Emsiz
Keyword(s):  
Haar Measure ◽  
Symplectic Group ◽  
Symmetric Functions ◽  
Hyperplane Arrangement ◽  
Cubature Formulas ◽  
Littlewood Polynomials ◽  
Orthogonality Relations ◽  
Unitary Ensemble ◽  
Determinantal Expression ◽  
Complex Hyperplane

Abstract Discrete orthogonality relations for Hall–Littlewood polynomials are employed so as to derive cubature rules for the integration of homogeneous symmetric functions with respect to the density of the circular unitary ensemble (which originates from the Haar measure on the special unitary group $SU(n;\mathbb{C})$). By passing to Macdonald’s hyperoctahedral Hall–Littlewood polynomials, we moreover find analogous cubature rules for the integration with respect to the density of the circular quaternion ensemble (which originates in turn from the Haar measure on the compact symplectic group $Sp (n;\mathbb{H})$). The cubature formulas under consideration are exact for a class of rational symmetric functions with simple poles supported on a prescribed complex hyperplane arrangement. In the planar situations (corresponding to $SU(3;\mathbb{C})$ and $Sp (2;\mathbb{H})$), a determinantal expression for the Christoffel weights enables us to write down compact cubature rules for the integration over the equilateral triangle and the isosceles right triangle, respectively.

scholarly journals Circle-valued Morse theory for complex hyperplane arrangements

2015 ◽  
Vol 27 (4) ◽  
Author(s):  
Toshitake Kohno ◽  
Andrei Pajitnov
Keyword(s):  
Morse Theory ◽  
Hyperplane Arrangement ◽  
Hyperplane Arrangements ◽  
Complex Hyperplane

AbstractLet 𝒜 be an essential complex hyperplane arrangement in

scholarly journals Tate properties, polynomial-count varieties, and monodromy of hyperplane arrangements

2012 ◽  
Vol 206 ◽  
pp. 75-97 ◽  
Author(s):  
Alexandru Dimca
Keyword(s):  
Hyperplane Arrangement ◽  
Hyperplane Arrangements ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Monodromy Operator ◽  
Low Dimensions ◽  
Milnor Fiber ◽  
Necessary And Sufficient ◽  
Complex Hyperplane

AbstractThe order of the Milnor fiber monodromy operator of a central hyperplane arrangement is shown to be combinatorially determined. In particular, a necessary and sufficient condition for the triviality of this monodromy operator is given.It is known that the complement of a complex hyperplane arrangement is cohomologically Tate and, if the arrangement is defined over ℚ, has polynomial count. We show that these properties hold for the corresponding Milnor fibers if the monodromy is trivial.We construct a hyperplane arrangement defined over ℚ, whose Milnor fiber has a nontrivial monodromy operator, is cohomologically Tate, and has no polynomial count. Such examples are shown not to exist in low dimensions.

Filling words in fundamental groups of Riemann surfaces

2013 ◽  
Vol 50 (1) ◽  
pp. 31-50
Author(s):  
C. Zhang
Keyword(s):  
Riemann Surface ◽  
Fundamental Group ◽  
Riemann Surfaces ◽  
Fundamental Groups ◽  
Closed Geodesics ◽  
New Words

The purpose of this article is to utilize some exiting words in the fundamental group of a Riemann surface to acquire new words that are represented by filling closed geodesics.

Representations of the fundamental group and the torsor of deformations. Global aspects

2017 ◽  
Author(s):  
Ahmed Abbes ◽  
Michel Gros
Keyword(s):  
Fundamental Group ◽  
Koszul Complex ◽  
Finiteness Conditions ◽  
Inverse Systems ◽  
Logarithmic Geometry

This chapter continues the construction and study of the p-adic Simpson correspondence and presents the global aspects of the theory of representations of the fundamental group and the torsor of deformations. After fixing the notation and general conventions, the chapter develops preliminaries and then introduces the results and complements on the notion of locally irreducible schemes. It also fixes the logarithmic geometry setting of the constructions and considers a number of results on the Koszul complex. Finally, it develops the formalism of additive categories up to isogeny and describes the inverse systems of a Faltings ringed topos, with a particular focus on the notion of adic modules and the finiteness conditions adapted to this setting. The chapter rounds up the discussion with sections on Higgs–Tate algebras and Dolbeault modules.

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