Analytic cycles on complex manifolds

Abstract

LET X be a complex manifold, Y a closed irreducible k-dimensional complex analytic subspace of X. Then Y defines or “carries” a 2k-dimensional integral homology class J’ of X, although the precise definition of y presents technical difficu1ties.S A finite formal linear combination 1 niYI with ni integers and Yi as above is called a complex analytic cycle, and the corresponding homology class c ni)vi is called a complex analytic homology class. If an integral cohomology class u corresponds under Poincart duality to a complex analytic homology class we shall say that u is a complex analytic cohomology class. The purpose of this paper is to show that a complex analytic cohomology class u satisfies certain topological conditions, independent of the complex structure of X. These conditions are that certain cohomology operations should vanish on U, for example Sq3u = 0: they are all torsion conditions. We also produce examples to show that these conditions are not vacuous even in the restricted classes of (a) Stein manifolds and (b) projective algebraic manifolds.