ÍâÍøÌìÌÃ

For a degenerating family of projective manifolds, it is ofÌýfundamental interest to study the asymptotic behavior of theÌýfiberwise mass of volume forms near singular fibers. We will discussÌýour main results where we determine the volume asymptoticsÌý(equivalently the asymptotics of $L^2$ metrics) in all baseÌýdimensions, which generalize numerous previous results in baseÌýdimension $1$. In the case of log Calabi-Yau fibrations, we establishÌýa metric version of the canonical bundle formula in algebraic geometry: the $L^2$ metric carries the singularity described by theÌýdiscriminant divisor and the moduli part line bundle has a singularÌýhermitian metric with vanishing Lelong numbers. As consequences, weÌýstrengthen the semipositivity theorems in algebraic geometry due toÌýFujita, Kawamata and others for log Calabi-Yau fibrations, giving anÌýentirely new simpler proof which does not use Hodge theory, i.e.Ìýdifficult results (e.g.ÌýCattani-Kaplan-Schmid) in the theory of variation of Hodge structure.

Ìý