## Advanced Studies in Pure Mathematics

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

### Resonance varieties and Dwyer–Fried invariants

#### 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