Local Cohomology of Analytic Spaces

Abstract

The purpose of this paper is to show that the local cohomology of a complex analytic space embedded in a complex manifold is a holonomic system of linear differential equations of infinite order and its holomorphic solution sheaves are a resolution of the constant sheaf $\mathbf{C}$ in this space which provides the Poincaré lemma. The proof relies on the theories of the $b$-function and holonomic systems due to M. Kashiwara ([2] and [3]) and A. Grothendieck's theorem on the De Rham cohomology of an algebraic variety ([1]). I am very much indebted to M. Kashiwara from whose papers I learned so much.