  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.
