On the rapid decay homology of F.Pham

TL;DR

This paper compares two versions of rapid decay homology groups associated with irregular connections on complex varieties, establishing a comparison theorem and applying it to construct bases for arrangements.

Contribution

It proves a comparison theorem between Hien's and Pham's rapid decay homology groups and applies this to arrangements, extending Sabbah's results.

Findings

Established a comparison theorem for rapid decay homology groups.

Constructed explicit bases for homologies of arrangements.

Extended the understanding of irregular connections and their homologies.

Abstract

In \cite{hien}, M. Hien introduced rapid decay homology group $\Homo^{rd}_{*}(U, (\nabla, E))$ associated to an irregular connection $(\nabla, E)$ on a smooth complex affine variety $U$, and showed that it is the dual group of the algebraic de Rham cohomology group $\Homo^*_{dR}(U,(\nabla^{\vee}, E^{\vee}))$. On the other hand, F. Pham has already introduced his version of rapid decay homology when $(\nabla, E)$ is the so-called elementary irregular connection (\cite{Sab}) in \cite{Pham}. In this report, we will state a comparison theorem of these homology groups and give an outline of its proof. This can be regarded as a homological counterpart of the result \cite{Sab} of C. Sabbah. As an application, we construct a basis of some rapid decay homologies associated to a hyperplane arrangement and hypersphere arrangement of Schl\"ofli type.