Relative cohomology for the sections of a complex of fine sheaves

Abstract

This is a summary of an article in preparation under the same title. In the Kinosaki AG Symposium 2017, I talked about relative Dolbeault cohomology and its application to the Sato hyperfunction theory, whose contents can be seen in [10], [18] and [19]. The idea is to represent the relative cohomology with coefficients in the sheaf of holomorphic forms by the relative Dolbeault cohomology, which enables us to express the relative cohomology in a simple explicit way. In this article we introduce a general theory of Cech cohomology for a complex of ne sheaves, which naturally leads to the notion of the relative cohomology of the associated complex of sections. In the case the complex gives a resolution of a certain sheaf, the relative cohomology of sections of the complex is canonically isomorphic with the relative cohomology of the sheaf. I would like to thank N. Honda for stimulating discussions while working on this and related subjects. Thanks are also due to T. Fujisawa and T. Terasoma for usuful conversations during the symposium.