  The goal of this papers is to extending to the complex analytic framework the
relative Kleiman duality for quasi coherent sheaves. Precisely, he show that
for any flat,locally projectivea and finitely presented morphism of schemes
$\pi:X\rightarrow S$ whose fibers are of pure dimension $n$, the functor ${\rm
I}\!{\rm R}^{n}\pi_{*}:{\rm Q}coh(X) \rightarrow {\rm Q}coh(S)$ admits a right
adjoint covariant functor (noted $\pi^{!}_{\mathcal K}$) inducing, for any
quasi-coherent sheaves ${\mathcal F}$ and ${\mathcal G}$ on $X$ and $S$
respectively, a relative duality isomorphism ${\rm I}\!{\rm H}om(X; {\mathcal
F},\pi^{!}_{\mathcal K}({\mathcal G}))\simeq {\rm I}\!{\rm H}om(S; {\rm
I}\!{\rm R}^{n}\pi_{*}{\mathcal F}, {\mathcal G})$ bifunctorial in ${\mathcal
F}$, ${\mathcal G}$ satisfying many nice functorial properties. Furthermore, he
shows that full duality is achieved if and only if $\pi$ is Cohen Macaulay
morphism. We show in the first part which concerns proper morphism of complex
spaces with constant fibers dimension that we have a similar duality in this
context with the same conclusion for the full duality. In the secod part, the
morphism are not necessarly proper but equidimensional or open with constant
fibers dimension. The situation is much more complicated because we must use a
specific analytic geometry tools. Infinite dimensional cohomology groups
considered as locally convex vectorial topological spaces are generally not
Hausdorff, the higher direct image with propre support ${\rm I}\!{\rm
R}^{n}\pi_{!}$ are generally never coherent. Despite all this problems, we are
able to give a semi-relative duality theorem as mentioned above and in a
certain sense.
