Homology of a local system on the complement of hyperplanes
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.
scientific article; zbMATH DE number 3988251
scientific article