  In this paper we study the singularities of the invariant metric of the
Poincar\'e bundle over a family of abelian varieties and their duals over a
base of arbitrary dimension. As an application of this study we prove the
effectiveness of the height jump divisors for families of pointed abelian
varieties. The effectiveness of the height jump divisor was conjectured by Hain
in the more general case of variations of polarized Hodge structures of weight
$-1$.
