On the exactness of a Kodaira-Spencer sequence on complex-spaces

Abstract

First we shall define ringed spaces and mappings of ringed spaces (see [2]). Let X be a topological space, let d = d ( X ) be a sheaf of a complex algebra on X and let all stalks dx, x~X, of X form a commutative and associative algebra with the unit lx, the mapping x -+ lx being continuous. Then the pair (X, d ) is called a ringed space and d is called a structure sheaf of X. Let (X, d ) and (Y, ~) be two ringed spaces. ~b=(~Oo, q)a) is called a mapping of ringed spaces of ( X , d ) into (Y,~) if ~oo: X---~Y and ~o1: XG~oN={(x,~r): x~X, aCNeo(x)}-~d are continuous mappings and (Pl maps each algebra (x, ~,o(~)) homomorphically into the algebra d x. ~b is called an isomorphism of ringed spaces if q)o and qh are topological mappings. Let D be a domain in C', let (9 = (9 (D) be the sheaf of germs of holomorphic functions over D, let A be an analytic set in D, let ~r be a coherent subsheaf of ideals of (9(D) whose locus (or the set of zeros) is A, and let ~4~ be the restriction of the quotient sheaf (9(D)/J to A. Then (A, J(r is said to be a ringed space. In the case J = J(A), the sheaf of ideals of all germs of holomorphic functions which vanish on A, we denote 2#(A) by (9 (A).