Finiteness of the space of n-cycles for a reduced (n -- 2)-concave complex space

TL;DR

This paper proves that in strongly (n-2)-concave complex spaces, the space of closed n-cycles forms a finite-dimensional reduced complex space that parametrizes analytic families of cycles.

Contribution

It establishes the finite-dimensional complex structure of the space of n-cycles in strongly (n-2)-concave complex spaces, extending understanding of their geometric properties.

Findings

The space of closed n-cycles is a reduced complex space.

This space is locally of finite dimension.

It parametrizes analytic families of n-cycles.

Abstract

We show that for n $\ge$ 2 the space of closed n-cycles in a strongly (n -- 2)-concave complex space has a natural structure of reduced complex space locally of finite dimension and represents the functor "analytic family of n-cycles" parametrized by Banach analytic sets.