This is a joint work with Dennis Eriksson and Gerard Freixas i Montplet. We consider a proper flat Kähler fibration between a complex manifold and a smooth complex curve, whose smooth fibers have trivial canonical bundle. Two types of metrics are natural in this setting, the L² metrics close to Hodge theory and the Quillen metrics with more topological features. I will explain how, extending a work of Fang–Lu–Yoshikawa in dimension 3 and inspired by Kazuya Kato, we build from those metrics on a special line bundle, the BCOV invariant, an invariant of the complex structure of each smooth fiber. I will describe its asymptotic along one parameter degenerations.