On the rapid decay homology of F.Pham (New development of microlocal analysis and singular perturbation theory)

Abstract

In [9], M. Hien introduced rapid decay homology group H_{*}^{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 H_{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 socalled elementary irregular connection ([20]) in [18]. 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 [20] of C. Sabbah. As an application, we construct a basis of some rapid decay homologies associated to a hyperplane arrangement and hypersphere arrangement of Schlöfli type.