  Calabi--Yau manifolds have risen to prominence in algebraic geometry, in part
because of mirror symmetry and enumerative geometry. After
Bershadsky--Cecotti--Ooguri--Vafa (BCOV), it is expected that genus 1 curve
counting on a Calabi--Yau manifold is related to a conjectured invariant, only
depending on the complex structure of the mirror, and built from Ray--Singer
holomorphic analytic torsions. To this end, extending work of
Fang--Lu--Yoshikawa in dimension 3, we introduce and study the BCOV invariant
of Calabi--Yau manifolds of arbitrary dimension. To determine it, knowledge of
its behaviour at the boundary of moduli spaces is imperative. We address this
problem by proving precise asymptotics along one-parameter degenerations, in
terms of topological data and intersection theory. Central to the approach are
new results on degenerations of $L^2$ metrics on Hodge bundles, combined with
information on the singularities of Quillen metrics in our previous work.
