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The main result of this article is the fact that the currents defined byLevin give a description of the polylogarithm of an abelian scheme at thetopological level. This result was a conjecture of Levin. This provides amethod to explicit the Eisenstein classes of an abelian scheme at thetopological level. These classes are of special interest since they have amotivic origin by a theorem of Kings. In a forthcoming article, we use the mainresult of this paper to prove that the Eisenstein classes of the universalabelian scheme over an Hilbert-Blumenthal variety degenerate at the boundary ofthe Baily-Borel compactification of the base in a special value of an$L$-function associated to the underlying totally real number field. As acorollary, we get a non vanishing result for some of these Eisenstein classesin this geometric situation.

Realisation de Hodge du polylogarithme d'un schema abelien