On the rapid decay homology of F.Pham

In \cite{hien}, M. Hien introduced rapid decay homology group ${\text{Homo}}_{\ast }^{rd}(U,(\mathrm{\nabla },E))$ associated to an irregular connection $(\mathrm{\nabla },E)$ on a smooth complex affine variety $U$, and showed that it is the dual group of the algebraic de Rham cohomology group ${\text{Homo}}_{dR}^{\ast }(U,({\mathrm{\nabla }}^{\vee },E^{\vee }))$. On the other hand, F. Pham has already introduced his version of rapid decay homology when $(\mathrm{\nabla },E)$ 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.