For a family of compact complex manifolds, the direct image of the canonical bundle (or Hodge bundle) admits a natural metric. Approaching singular fibers this metric degenerates, and we provide explicit expressions for the dominant terms in the Calabi-Yau case. These are described in terms of topological invariants coming from vanishing cycles or limit Hodgestructures. I will also discuss applications to BCOV-torsion or metrics, whose motivation originally came from mirror symmetry. This is joint work with Gerard Freixas and Christophe Mourougane.