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