Polylogarithmic Extensions on Mixed Shimura varieties, by Joerg Wildeshaus

This paper is a slightly revised version of the author's thesis (November 1993) and is identical to the Heft 12 of the Schriftenreihe des Mathematischen Instituts Muenster. It aims at a generalization on the level of realizations, i.e., "ex K-theory", of the classical and elliptic polylogarithms (as defined and studied by Beilinson, Deligne and Levin) to the context of mixed Shimura varieties, where phaenomena like the rigidity principle or the splitting of the "logarithmic sheaf" over torsion sections also occur.

Chapter I (chapter1.dvi) gives partial results on what we call the "generic relatively unipotent sheaf", which is defined for any sufficiently nice morphism with a section of schemes over C (in the Hodge theoretic context) or a number field (in the context of l-adic sheaves, or systems of smooth sheaves). In the Hodge theoretic context, the generic sheaf is a relative version of what is called the "canonical variation with basepoint" in the work of Hain and Zucker. Chapter II (chapter2.dvi) starts with a discussion of the canonical construction of mixed sheaves on mixed Shimura varieties. The generic sheaf of chapter I is renamed as the "logarithmic sheaf". We then give the general formalism of the construction of polylogarithmic extensions (indeed, in the higher dimensional case, these extensions won't be one-extensions, hence can't be thought of as framed sheaves). In the final chapter III (chapter3.dvi), we convince ourselves that what we have done really constitutes a generalization of the two constructions which are already existent. The paragraph about the classical polylogarithm doesn't contain any results that cannot be found in the work of Beilinson and Deligne. Still, we find it worth to reprove them in the context of mixed Shimura varieties. The final paragraph gives a description of the Hodge theoretic incarnation of the elliptic polylogarithm which we like to think of as new.

As mentioned in the introduction (Intro.dvi), considerable progress has been made since the thesis was completed. It concerns in particular the results of chapter I, so we added its revised version (preprint.dvi) to our collection of files. It will itself undergo some sort of revision since some of the results (e.g. concerning the de Rham version of the generic sheaf) seem to have been obtained before by Z. Wojtkowiak, a fact that is not yet mentioned in the preprint. Once a final version of preprint.dvi is available, it will be added to the archive. The other two chapters have also been revised, and the results will be available in the near future, either in this or the algebraic geometry archive (alg-geom@edu.duke.math.publications).

This thesis has been removed, as it has been republished in the form of the following five papers.

• 0026.bib (248 bytes)