On the Blowing Down of Analytic Spaces

Abstract

Let X be a complex space and A be a subspace of X. Suppose there exists a proper surjective morphism/: A-+A', where A' is another complex space. We say that X can be blown down along /, if the following conditions are satisfied; there exists a complex space X' containing A' as a subspace, and a proper surjective morphism /': X—*X' such that i) f'(A)=A and f\A coincides with/, and ii) / gives an isomorphism of X—A and X'—A'. In this case we say that (-<¥',/') is the blowing down of X along/. Now given a triple (X, A,/} as above, the problem of finding conditions for blowing down X along / has been investigated by many people from various points of view [1] [3] [7] [8] [13] [17] [18] [19] [21] [22] [23] [26]. In this paper we give one sufficient condition for blowing down X under the assumption that A is an effective Cartier divisor on X (Theorem 2). This theorem has been proved by Artin [1] in the category of algebraic spaces. But our method here is a direct generalization of that of [23], and uses a cohomology vanishing theorem for weakly 1-complete complex spaces, which generalizes a similar theojem of Nakano [24]. (Theorem N' in § 1). Then in §2, we treat the local version of the problem and obtain Theorem 1, the proof of which is the main part of this paper. Next in §3, we patch together the local blowing downs and obtain a global one. Geometrically, the condition of the theorem says that the normal bundle of A in X, when restricted to each fiber of/, is sufficiently negative. In §3 we also show by an example that this 'sufficient' negativity condition is