Multivariable Alexander invariants of hypersurface complements

Dimca, Alexandru; Maxim, Laurenţiu

doi:10.1090/s0002-9947-07-04241-9

Cited by 32 publications

(42 citation statements)

References 45 publications

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“…In this sense this result is in the same vein as the results of Libgober [Lib82] and Cogolludo-Florens [CF07], but see also [Lib94,DM07,Ma06] for similar results in the higher-dimensional case. …”

Section: Introductionsupporting

confidence: 85%

L 2-Betti numbers of plane algebraic curves

Friedl¹,

Leidy²,

Maxim³

2009

Michigan Math. J.

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Section: Introductionsupporting

confidence: 85%

L 2-Betti numbers of plane algebraic curves

Friedl¹,

Leidy²,

Maxim³

2009

Michigan Math. J.

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“…As it was already observed in a sequence of papers, e.g., see [DL06,DM07,Ma06], such hypersurfaces behave much like weighted homogeneous hypersurfaces up to homological degree n − 1.…”

Section: Betti Numbers Of Hypersurface Complementssupporting

confidence: 58%

“…As it was already observed in several recent papers, e.g., see [DL06,DM07,Ma06], hypersurfaces in general position at infinity behave much like weighted homogeneous hypersurfaces up to homological degree n − 1; see Prop.3.1 for a computation of L 2 -Betti numbers of weighted homogeneous hypersurface complements. The above Theorem 1.3 comes as a confirmation of this philosophy.…”

Section: Introductionsupporting

confidence: 52%

L 2-Betti Numbers of Hypersurface Complements

Maxim

2013

International Mathematics Research Notices

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Abstract. In [DJL07] it was shown that if A is an affine hyperplane arrangement in C n , then at most one of the L 2 -Betti numbers b(2) i (C n \ A, id) is non-zero. In this note we prove an analogous statement for complements of complex affine hypersurfaces in general position at infinity. Furthermore, we recast and extend to this higher-dimensional setting results of [FLM09,LM06] about L 2 -Betti numbers of plane curve complements. IntroductionLet M be any topological space and α : π 1 (M ) → Γ an epimorphism to a group Γ (all groups are assumed countable). Then for i ∈ N ∪ {0} we can consider theWe recall the definition and some of the most important properties of L 2 -Betti numbers in Section 2. Let X ⊂ C n (n ≥ 2) be a reduced affine hypersurface defined by a polynomial equation f = f 1 · · · f s = 0, where f i are the irreducible factors of f . Denote by X i := {f i = 0}, i = 1, · · · , s, the irreducible components of X, and letbe the hypersurface complement. Then M X has the homotopy type of a finite CW complex of dimension n. It is well-known that H 1 (M X ; Z) is a free abelian group generated by the meridian loops γ i about the non-singular part of each irreducible component X i of X. Throughout the paper we denote by φ the map π 1 (M X ) → Z given by sending each meridian γ i to 1. This is the same map as the homomorphism f * : π 1 (M X ) → π 1 (C * ) = Z induced by f . We also refer to φ as the total linking number homomorphism. We call an epimorphism α : π 1 (M X ) → Γ admissible if the total linking number homomorphism φ factors through α.The main result of this note is the following "nonresonance-type" theorem.Theorem 1.1. Let X ⊂ C n be a reduced affine hypersurface in general position at infinity, i.e., whose projective completion intersects the hyperplane at infinity transversely. If α : π 1 (M X ) → Γ is an admissible epimorphism, then the L 2 -Betti numbers of the complement M X are computed by:

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“…The second-named author ( [Max06]) extended these divisibility results to the case of hypersurfaces with arbitrary singularities and in general position at infinity. Furthermore, this global-tolocal approach was used in [DM07] to show that such divisibility results also hold for certain multivariable Alexander-type invariants, called the Alexander varieties (or support loci) of the hypersurface complement.…”

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confidence: 99%

Characteristic varieties of hypersurface complements

Yong-qiang

Maxim

2017

Advances in Mathematics

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Abstract. We give divisibility results for the (global) characteristic varieties of hypersurface complements expressed in terms of the local characteristic varieties at points along one of the irreducible components of the hypersurface. As an application, we recast old and obtain new finiteness and divisibility results for the classical (infinite cyclic) Alexander modules of complex hypersurface complements. Moreover, for the special case of hyperplane arrangements, we translate our divisibility results for characteristic varieties in terms of the corresponding resonance varieties.

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Multivariable Alexander invariants of hypersurface complements

Cited by 32 publications

References 45 publications

L 2-Betti numbers of plane algebraic curves

L 2-Betti numbers of plane algebraic curves

L 2-Betti Numbers of Hypersurface Complements

Characteristic varieties of hypersurface complements

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