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The Hodge realization of the polylogarithm of an Abelian scheme is an extension of Hodge modules. In [A. Levin, Polylogarithmic currents on abelian varieties, in: A. Reznikov, N. Schappacher (Eds.), Regulators in Analysis, Geometry and Number Theory, in: Progr. Math., vol. 171, Birkhäuser, 2000, pp. 207–229], Levin constructs some currents (generalized Eisenstein series) and conjectures that they describe the polylogarithmic extension. Our main result (Thm 3.1 and Cor 3.2) is a proof of this conjecture. This provides a tool to study the Eisenstein classes (see Section 4), which have a motivic origin (see [G. Kings, K -theory elements for the polylogarithm of abelian schemes, J. Reine Angew. Math. 517 (1999) 103–116]); an example of its application will be given in a second Note. To cite this article: D. Blottière, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

Réalisation de Hodge du polylogarithme d'un schéma abélien