A global morphism from the Douady space to the cycle space

Abstract

We establish, for any given complex space $M$, a global morphism from the reduction of its Douady space to its cycle space. This morphism is an extension of the morphism defined in [1] from the subspace of the Douady space formed by all pure dimensional subspaces of $M$ to the cycle space of $M$. In the case where $M$ is projective this morphism is the classical morphism from the Hilbert scheme of $M$ to the Chow scheme of $M$.