Homology of a local system on the complement of hyperplanes
type: Item
instance of: scholarly article
title: Homology of a local system on the complement of hyperplanes
zbMATH Open document ID: 0611.55005
author: Toshitake Kohno
published in: Proceedings of the Japan Academy. Series A
publication date: 1986-01-01T00:00:00Z
review text: Let \(\Omega =\Sigma P_ j d \log f_ j\) denote a connection composed of endomorphisms \(P_ j\) of a complex vector space and of linear functions \(f_ j\) on \({\mathbb{C}}^ n\). The solutions of the system of differential equations \(dY+\Omega Y=0\) define a local system \({\mathcal L}\) on the complement X of the affine hyperplanes \(A_ j\) defined by \(f_ j\). The homology \(H_ k(X, {\mathcal L})\) vanishes for \(k\neq n\) if the eigenvalues of \(P_ j\) and certain sums \(P_{j_ 1}+...+P_{j_ q}\) are not entire. Furthermore for real \(f_ j's\) a base of \(H_ n(X, {\mathcal L})\) is given in terms of the relative compact chambers of the real part of X. The proof is rather short: After a monoidal transformation along \(A_{j_ 1}\cap...\cap A_{j_ q}\) the Picard-Lefschetz formulas are applied.
Mathematics Subject Classification ID: 55N25
Mathematics Subject Classification ID: 34M99
Mathematics Subject Classification ID: 32C35
Mathematics Subject Classification ID: 14E20
Mathematics Subject Classification ID: 14F25
zbMATH DE Number: 3988251
zbMATH Keywords: complex differential equations
zbMATH Keywords: homology with local coefficients
zbMATH Keywords: complements of hyperplanes
zbMATH Keywords: monoidal transformation
zbMATH Keywords: Picard-Lefschetz formulas
MaRDI profile type: Publication
full work available at URL: pjaa.62.144
OpenAlex ID: W2036024677
cites work: Q5622357
cites work: On vanishing of cohomology attached to certain many valued meromorphic functions
cites work: Gauss-Manin connection of integral of difference products
cites work: Equations différentielles à points singuliers réguliers
cites work: Monodromy of hypergeometric functions and non-lattice integral monodromy
cites work: Q4062377
cites work: Q3342869
cites work: On the holonomy Lie algebra and the nilpotent completion of the fundamental group of the complement of hypersurfaces
cites work: Série de Poincaré-Koszul associée aux groupes de tresses pures
cites work: Facing up to arrangements: face-count formulas for partitions of space by hyperplanes
DOI: 10.3792/PJAA.62.144
description: scientific article; zbMATH DE number 3988251