4,898 research outputs found
On the cohomology of discriminantal arrangements and Orlik-Solomon algebras
We relate the cohomology of the Orlik-Solomon algebra of a discriminantal
arrangement to the local system cohomology of the complement. The Orlik-Solomon
algebra of such an arrangement (viewed as a complex) is shown to be a linear
approximation of a complex arising from the fundamental group of the
complement, the cohomology of which is isomorphic to that of the complement
with coefficients in an arbitrary complex rank one local system. We also
establish the relationship between the cohomology support loci of the
complement of a discriminantal arrangement and the resonant varieties of its
Orlik-Solomon algebra.Comment: LaTeX2e, 16 pages, to appear in Singularities and Arrangements,
Sapporo-Tokyo 1998, Advanced Studies in Pure Mathematic
Resonance of basis-conjugating automorphism groups
We determine the structure of the first resonance variety of the cohomology
ring of the group of automorphisms of a finitely generated free group which act
by conjugation on a given basis.Comment: 7 page
Arrangements and local systems
We use stratified Morse theory to construct a complex to compute the
cohomology of the complement of a hyperplane arrangement with coefficients in a
complex rank one local system. The linearization of this complex is shown to be
the Orlik-Solomon algebra with the connection operator. Using this result, we
establish the relationship between the cohomology support loci of the
complement and the resonance varieties of the Orlik-Solomon algebra for any
arrangement, and show that the latter are unions of subspace arrangements in
general, resolving a conjecture of Falk. We also obtain lower bounds for the
local system Betti numbers in terms of those of the Orlik-Solomon algebra,
recovering a result of Libgober and Yuzvinsky. For certain local systems, our
results provide new combinatorial upper bounds on the local system Betti
numbers. These upper bounds enable us to prove that in non-resonant systems the
cohomology is concentrated in the top dimension, without using resolution of
singularities.Comment: LaTeX, 14 page
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