Strongly quasi-proper maps and the f-flattening theorem

TL;DR

This paper advances the understanding of strongly quasi-proper maps by providing a precise semi-proper direct image theorem, characterizing these maps, and offering a simplified proof of Mathieu's flattening theorem.

Contribution

It completes previous results, characterizes strongly quasi-proper maps, and simplifies the proof of Mathieu's flattening theorem in complex geometry.

Findings

Established a strong semi-proper direct image theorem.

Characterized strongly quasi-proper maps as holomorphic surjective maps with meromorphic fibers.

Provided a simplified proof of Mathieu's flattening theorem.

Abstract

We complete and precise the results of [B.13] and we prove a strong version of the semi-proper direct image theorem with values in the space C f n (M) of finite type closed n--cycles in a complex space M. We describe the strongly quasi-proper maps as the class of holomorphic surjective maps which admit a meromorphic family of fibers and we prove stability properties of this class. In the Appendix we give a direct and short proof of D. Mathieu's flattening theorem (see [M.00]) for a strongly quasi-proper map which is easier and more accessible.