A general extension theorem for cohomology classes on non reduced analytic spaces

The main purpose of this paper is to generalize the celebrated L${}^2$ extension theorem of Ohsawa-Takegoshi in several directions : the holomorphic sections to extend are taken in a possibly singular hermitian line bundle, the subvariety from which the extension is performed may be non reduced, the ambient manifold is K{"a}hler and holomorphically convex, but not necessarily compact.