Vanishing and bases for cohomology of partially trivial local systems on hyperplane arrangements

Kawahara, Yukihito

doi:10.1090/s0002-9939-05-07745-2

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Cited by 2 publications

(3 citation statements)

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“…Note that λ H ∈ Z ≥0 for all hyperplanes in A. We can generalize this theorem in the case that there is H ∈ A with λ H = 0, by using [Ka2].…”

Section: Generalizationmentioning

confidence: 96%

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Non-vanishing of the twisted cohomology on the complement of hypersurfaces

Kawahara

2008

Proc. Amer. Math. Soc.

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Abstract. Generically, the cohomology with coefficients in a local system of rank one on the complement in P n of the union of a finite number of hypersurfaces vanishes except in the highest dimension. We study the non-generic case, in which the cohomology in other dimensions does not vanish. When the hypersurfaces are hyperplanes, many examples of this kind are known. In this paper, we consider the case in which the hypersurfaces need not be hyperplanes. We prove that the hypersurfaces given by some particular linear systems have non-vanishing local system cohomologies.

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“…Note that λ H ∈ Z ≥0 for all hyperplanes in A. We can generalize this theorem in the case that there is H ∈ A with λ H = 0, by using [Ka2].…”

Section: Generalizationmentioning

confidence: 96%

“…We can generalize this theorem to the case in which there is a hyperplane H ∈ A with λ H = 0 by using [Ka2].…”

Section: Remarkmentioning

confidence: 99%

See 1 more Smart Citation

Non-vanishing of the twisted cohomology on the complement of hypersurfaces

Kawahara

2008

Proc. Amer. Math. Soc.

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Bases for certain cohomology representations of the symmetric group

Henderson

2006

J Algebr Comb

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Abstract. We give a combinatorial description (including explicit differential-form bases) for the cohomology groups of the space of n distinct nonzero complex numbers, with coefficients in rank-one local systems which are of finite monodromy around the coordinate hyperplanes and trivial monodromy around all other hyperplanes. In the case where the local system is equivariant for the symmetric group, we write the cohomology groups as direct sums of inductions of one-dimensional characters of subgroups. This relies on an equivariant description of the Orlik-Solomon algebras of full monomial reflection groups (wreath products of the symmetric group with a cyclic group). The combinatorial models involved are certain representations of these wreath products which possess bases indexed by labelled trees.

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Vanishing and bases for cohomology of partially trivial local systems on hyperplane arrangements

Abstract: Abstract. In this paper we prove a vanishing theorem and construct bases for the cohomology of partially trivial local systems on complements of hyperplane arrangements. As a result, we obtain a non-resonance condition for partially trivial local systems.

Cited by 2 publications

References 20 publications

Non-vanishing of the twisted cohomology on the complement of hypersurfaces

Non-vanishing of the twisted cohomology on the complement of hypersurfaces

Bases for certain cohomology representations of the symmetric group

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