# Canonical bundle formula and degenerating families of volume forms

# Canonical bundle formula and degenerating families of volume forms

For a degenerating family of projective manifolds, it is of fundamental interest to study the asymptotic behavior of the fiberwise mass of volume forms near singular fibers. We will discuss our main results where we determine the volume asymptotics (equivalently the asymptotics of $L^2$ metrics) in all base dimensions, which generalize numerous previous results in base dimension $1$. In the case of log Calabi-Yau fibrations, we establish a metric version of the canonical bundle formula in algebraic geometry: the $L^2$ metric carries the singularity described by the discriminant divisor and the moduli part line bundle has a singular hermitian metric with vanishing Lelong numbers. As consequences, we strengthen the semipositivity theorems in algebraic geometry due to Fujita, Kawamata and others for log Calabi-Yau fibrations, giving an entirely new simpler proof which does not use Hodge theory, i.e. difficult results (e.g. Cattani-Kaplan-Schmid) in the theory of variation of Hodge structure.