# Platitude géométrique et classes fondamentales relatives pondérées I

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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 analyticallyresp. 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.