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 L2 metrics on Hodge bundles, combined with
information on the singularities of Quillen metrics in our previous work.Comment: Minor revision. Mainly restructure of the text, minor improvements
and corrections. Added information about subdominant terms of L2-norm