During the preparation of this text we learned that conjectures similar to ours were proposed independently by M. Gross and P. Wilson (see [GW]). A remarkable achievement in [GW] consists of the verification of conjectures in the case of degenerating $K3$ surfaces, together with a precise description of the behavior of metrics near singular fibers. Also, in a recent preprint [Le] mirror symmetry was discussed from a similar point of view. In the main body of the present paper we will consider degenerations of complex abelian varieties. In this case the conjectures obviously hold.