Advanced Studies in Pure Mathematics

Resonance varieties and Dwyer–Fried invariants

Alexander I. Suciu

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Abstract

The Dwyer–Fried invariants of a finite cell complex $X$ are the subsets $\Omega^i_r (X)$ of the Grassmannian of $r$-planes in $H^1 (X, \mathbb{Q})$ which parametrize the regular $\mathbb{Z}^r$-covers of $X$ having finite Betti numbers up to degree $i$. In previous work, we showed that each $\Omega$-invariant is contained in the complement of a union of Schubert varieties associated to a certain subspace arrangement in $H^1 (X, \mathbb{Q})$. Here, we identify a class of spaces for which this inclusion holds as equality. For such “straight” spaces $X$, all the data required to compute the $\Omega$-invariants can be extracted from the resonance varieties associated to the cohomology ring $H^{\ast} (X, \mathbb{Q})$. In general, though, translated components in the characteristic varieties affect the answer.

Article information

Source
Arrangements of Hyperplanes — Sapporo 2009, H. Terao and S. Yuzvinsky, eds. (Tokyo: Mathematical Society of Japan, 2012), 359-398

Dates
Received: 20 July 2010
Revised: 12 December 2010
First available in Project Euclid: 24 November 2018

Permanent link to this document
https://projecteuclid.org/ euclid.aspm/1543085015

Digital Object Identifier
doi:10.2969/aspm/06210359

Mathematical Reviews number (MathSciNet)
MR2933803

Zentralblatt MATH identifier
1273.14110

Subjects
Primary: 20J05: Homological methods in group theory 55N25: Homology with local coefficients, equivariant cohomology
Secondary: 14F35: Homotopy theory; fundamental groups [See also 14H30] 32S22: Relations with arrangements of hyperplanes [See also 52C35] 55R80: Discriminantal varieties, configuration spaces 57M07: Topological methods in group theory

Keywords
Free abelian cover characteristic variety resonance variety tangent cone Dwyer–Fried set special Schubert variety toric complex Kähler manifold hyperplane arrangement

Citation

Suciu, Alexander I. Resonance varieties and Dwyer–Fried invariants. Arrangements of Hyperplanes — Sapporo 2009, 359--398, Mathematical Society of Japan, Tokyo, Japan, 2012. doi:10.2969/aspm/06210359. https://projecteuclid.org/euclid.aspm/1543085015


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