Platitude g\'eom\'etrique et classes fondamentales relatives pond\'er\'ees I

TL;DR

This paper introduces a new geometric flatness concept for morphisms between complex spaces, linking it to fundamental class morphisms and extending previous characterizations using sheaves of meromorphic forms.

Contribution

It defines analytically and continuously geometrically flat morphisms via cycles and fundamental class morphisms, generalizing prior results and providing new characterizations.

Findings

Characterization of geometrically flat morphisms via fundamental class morphisms

Extension of previous flatness criteria to more general settings

Connection with sheaves of meromorphic relative forms

Abstract

Let $X$ and $S$ be complex spaces with $X$ countable at infinity and $S$ reduced locally pure dimensional. Let $\pi:X\to S$ be an universally-$n$-equidimensional morphism (i.e open with constant pure $n$-dimensional fibers). If there is a cycle $\goth{X}$ of $X\times S$ such that, his support coincide fiberwise set-theorically with the fibers of $\pi$ and endowed this with a good multiplicities in such a way that $(\pi^{-1}(s))_{s\in S}$ becomes a local analytic (resp. continuous) family of cycles in the sense of [B.M], $\pi$ is called analytically(resp. continuously) geometrically flat according to the weight $\goth{X}$. One of many results obtained in this work say that an universally-$n$-equidimensional morphism is analytically geometrically flat if and only if admit a weighted relative fundamental class morphism satisfies many nice functorial properties which giving, for a finite Tor-dimensional morphism or in the embedding case, the relative fundamental class of Angeniol-Elzein [E.A] or Barlet [B4]. From this, we deduce the generalization result [Ke] and nice characterization of analytically geometrically flatness by the Kunz-Waldi sheaf of regular meromorphic relative forms.