Twisted de Rham cohomology groups of logarithmic forms
For an \(l\)-dimensional complex affine space \(V\) with coordinate ring \(S\), let \(\Omega^{p}[V]\) be the \(S\)-module of global regular \(p\)-forms on \(V\), \(\Omega^{p}(V) \) the space of global rational forms. The module \(\Omega^{p}(\mathcal G)\) of logarithmic \(p\)-forms with poles along \(\mathcal G\) is defined as \[ \Omega^{p}(\mathcal G)=\{\omega \in \Omega^{p}(V) \mid Q\omega \in \Omega^{p} [V],\;Q(d\omega) \in \Omega^{p+1}[V] \} \] where \(\mathcal G\) is an affine \(l\)-arrangement with \(n\) hyperplanes \(H_{i}\) in general position, \(Q=\prod{\alpha_{i}}\), \(\alpha_{i}\) is a linear polynomial and \(H_{i}= \ker(\alpha_{i})\). Taking a polynomial \(f\) of degree \( q+1>0 \), the operator \[ \nabla_{f}=d+ \sum{\lambda_{i} \frac{d \alpha_{j}}{\alpha_{j}} \wedge} + df \wedge \] gives rise to a cochain complex \((\Omega^{\bullet}(\mathcal{G}),\nabla_{f})\). The main result of this paper is the computation of the cohomology groups of the twisted de Rham complex of the operator \(\nabla_{f}\), including vanishing, dimension, basis and so on.
scientific article; zbMATH DE number 1044218
scientific article