Resonance varieties and Dwyer-Fried invariants

Details Resonance varieties and Dwyer-Fried invariants Resonance varieties and Dwyer-Fried invariants

2011-11-19

arxiv.org/abs/1111.4534v1

Mathematics

Algebraic Geometry

39 pages; to appear in "Arrangements of Hyperplanes - Sapporo 2009," Advanced Studies in Pure Mathematics

Scientific paper

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,\Q) which parametrize the regular \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,\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^*(X,\Q). In general, though, translated components in the characteristic varieties affect the answer.

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