Characteristic varieties of quasi-projective manifolds and orbifolds

Bartolo, Enrique Artal; Cogolludo-Agustín, José Ignacio; Matei, Daniel

doi:10.2140/gt.2013.17.273

Cited by 37 publications

(84 citation statements)

References 39 publications

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“…The topological approach to Problem traces its origins to the work of Cohen and Suciu on Milnor fibrations and characteristic varieties of arrangements, which builds in turn on Arapura's theory of characteristic varieties of quasi‐projective manifolds. This theory, as refined in , provides a geometric interpretation of these topologically defined varieties in terms of (orbifold) pencils.…”

Section: Discussionmentioning

confidence: 98%

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The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy

Papadima

Suciu

2017

Proc. London Math. Soc.

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A central question in arrangement theory is to determine whether the characteristic polynomial Δq of the algebraic monodromy acting on the homology group Hqfalse(F(A),double-struckCfalse) of the Milnor fiber of a complex hyperplane arrangement scriptA is determined by the intersection lattice L(A). Under simple combinatorial conditions, we show that the multiplicities of the factors of Δ1 corresponding to certain eigenvalues of order a power of a prime p are equal to the Aomoto–Betti numbers βpfalse(scriptAfalse), which in turn are extracted from L(A). When scriptA defines an arrangement of projective lines with only double and triple points, this leads to a combinatorial formula for the algebraic monodromy. To obtain these results, we relate nets on the underlying matroid of scriptA to resonance varieties in positive characteristic. Using modular invariants of nets, we find a new realizability obstruction (over double-struckC) for matroids, and estimate the number of essential components in the first complex resonance variety of scriptA. Our approach also reveals a rather unexpected connection of modular resonance with the geometry of prefixSL2false(double-struckCfalse)‐representation varieties, which are governed by the Maurer–Cartan equation.

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Section: Discussionmentioning

confidence: 98%

“…Before proceeding, we need to recall a result of Artal Bartolo, Cogolludo‐Agustín, and Matei [, Proposition 6.9].…”

Section: Characteristic Varieties and The Milnor Fibrationmentioning

confidence: 99%

The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy

Papadima

Suciu

2017

Proc. London Math. Soc.

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“…Results by Arapura [4], as strengthened in [7,15], put strong restrictions on the characteristic varieties of both Kähler manifolds and smooth, quasi-projective varieties. Following the approach from [14], we use those restrictions to conclude that the respective Alexander polynomials are 'thin': their Newton polytopes have dimension at most 1.…”

Section: 4mentioning

confidence: 99%

Kähler groups, quasi-projective groups and 3-manifold groups

Friedl

Suciu

2013

Journal of the London Mathematical Society

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We prove two results relating 3‐manifold groups to fundamental groups occurring in complex geometry. Let N be a compact, connected, orientable 3‐manifold. If N has non‐empty, toroidal boundary and π1(N) is a Kähler group, then N is the product of a torus with an interval. On the other hand, if N has either empty or toroidal boundary, and π1(N) is a quasi‐projective group, then all the prime components of N are graph manifolds.

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“…[9], Simpson-Corlette [12], Delzant [13] and ourselves [4], among others. Except in Campana's work, the relationship comes from the following fact: given a smooth variety (projective, quasiprojective or Kähler) the positive-dimensional components of the characteristic varieties can be obtained as pull-back by mappings whose targets are orbifolds.…”

Section: Orbifolds and Characteristic Varietiesmentioning

confidence: 99%

“…Our second purpose (see §3) is to extend two classical results regarding the variety of characters on smooth quasi-projective fundamental groups (due to Arapura [1] and the authors [4]) and normal-crossing compact Kähler orbiface groups (due to Campana [9]) to the general case of normal-crossing quasi-projective orbifold groups.…”

Section: Introductionmentioning

confidence: 99%

Orbifold Groups, Quasi-Projectivity and Covers

Bartolo¹,

Cogolludo-Agustín²,

Matei³

2012

jsing

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Abstract. We discuss properties of complex algebraic orbifold groups, their characteristic varieties, and their abelian covers. In particular, we deal with the question of (quasi)-projectivity of orbifold groups. We also prove a structure theorem for the variety of characters of normalcrossing quasi-projective orbifold groups. Finally, we extend Sakuma's formula for the first Betti number of abelian covers of orbifold fundamental groups. Several examples are presented, including a compact orbifold group which is not projective and a Zariski pair of plane curves in P 2 that can be told by considering an unbranched cover of P 2 with an orbifold structure.

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Characteristic varieties of quasi-projective manifolds and orbifolds

Cited by 37 publications

References 39 publications

The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy

The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy

Kähler groups, quasi-projective groups and 3-manifold groups

Orbifold Groups, Quasi-Projectivity and Covers

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