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

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.