  In this paper, we present an explicit formula for the Baer invariant of a
finitely generated abelian group with respect to the variety of polynilpotent
groups of class row $(c_1,...,c_t)$, ${\cal N}_{c_1,...,c_t}$. In particular,
one can obtain an explicit structure of the $\ell$-solvable multiplier (the
Baer invariant with respect to the vaiety of solvable groups of length at most
$\ell\geq 1,\ {\cal S}_{\ell}$.) of a finitely generated abelian group.
