This paper is the revised version of chapter I of Polylogarithmic Extensions on Mixed Shimura varieties. Its subject is the so-called "generic relatively unipotent sheaf", which is defined for any sufficiently nice morphism with a section over C (in the Hodge theoretic context), over any subfield of C (in the de Rham context) or number field (in the context of l-adic sheaves, or mixed systems of smooth sheaves). In the Hodge theoretic language, the generic sheaf is a relative version of what Hain and Zucker call the "canonical variation with basepoint".

The emphasis of the paper is the formulation in Tannakian terms of the classification theorem for relatively unipotent sheaves, and its consequences. In particular, we obtain a characterization of the generic sheaf by a universal property, which turns out to be a very powerful tool indeed. Its applications include the following:

i) The universal property of the Hodge version of the generic sheaf allows to conclude that its Hodge filtration descends to a subfield of C if the geometric situation descends. (This is how the de Rham version of the generic sheaf is constructed.)

ii) In either setting, the universal property facilitates considerably the study of the functor "canonical construction" of sheaves on mixed Shimura varieties from algebraic representations of the group underlying the Shimura data. As a consequence, it can e.g. be shown that the variations of Hodge structure obtained by the canonical construction are all admissible in the sense of Kashiwara.

iii) Since the universal property can be thought of as a rigidity statement on the mixed structure on the completed group ring of the relative fundamental group, it is of considerable interest for constructive purposes.

This paper is now available in Springer Lecture Notes in Mathematics, number 1650.

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Joerg Wildeshaus <wildesh@math.uni-muenster.de>