Showing papers on "Complex dimension published in 1975"

Neighborhoods of a Compact Non-Singular Algebraic Curve Imbedded in a 2-Dimensional Complex Manifold

Abstract: Let C be a compact non-singular algebraic curve imbedded in a 2-dimensional complex manifold S. In this paper we consider the following problem: Under what conditions does there exist a non-constant holomorphic function defined on a small neighborhood of Cl The signature of the normal bundle Nc is not sufficient to solve our problem (see, Table in § 6 and Theorem 2 in § 5). Hence we have to introduce the concept of a regularly half pseudoconvex neighborhood system of C (for definition, see (1.1)). Then the necessary and sufficient condition is given in the following

A one dimensional manifold is of cohomological dimension 2

Abstract: G. Bredon defines the cohomological Dimension of a topological space X to be the supremum of all cohomological +t-dimensions of X, where k varies over the entire families of supports on X. He has proved that if X is a topological n-manifold then the cohomological Dimension of X is n or n + 1. He was not able to decide which one it is, even for a space as simple as the real line. The objective of this paper is to solve his problem for n = 1. In particular, we have shown that the cohomological Dimension of the real line is 2. Let ? be a family of supports on a topological space X. Godement [2] defines the f-dimension (dim?, (X)) of X to be the largest integer n (or W) if there is a sheaf a of abelian groups on X for which the Grothendieck cohomology [3] group Hn,(X, () 0 0. If ? runs over all those paracompactifying families of supports on X whose extents equal X, then dim?,(X) is independent of ? and is called the cohomological dimension (dim(X)) of X. The class of all such spaces for which dim(X) has a meaning is large enough to include all locally paracompact Hausdorff spaces. The dim(X) of a n-manifold X is n and is dominated by the covering dimension [4] for paracompact Hausdorff X. Quite naturally, Bredon [1] defines the cohomological Dimension (Dim(X)) of X to be the Sup?,}dim?,(X)l for all families of supports ? on X. Among other things, he proves that for a topological n-manifold X, Dim(X) = n or n + 1 leaving it open which one it is. The present objective is to indicate its pathological or nonpathological nature by proving the following Theorem. Let X be a one-dimensional topological manifold. Then Dim(X) is two. Recall that if a is a sheaf of abelian groups on a space X and as(U) denotes the group of all serrations of a on U, then the sheaf generated by the presheaf U (s(U)/A(U) on X is denoted by Z1(X, a) and Dim(X) < 1 if and only if 91(X, a) is flabby for every sheaf a on X [1, p. 110]. Consider R as imbedded in the one-dimensional manifold X. Then the proof of the theorem follows from Received by the editors July 16, 1974. AMS (MOS) subject classifications (1970). Primary 55B30; Secondary 55C10.