Dano Kim (Seoul National University)

Title: Canonical bundle formula and degenerating families of volume forms

Abstract: For a degenerating family of projective manifolds, it is of fundamental interest to study the asymptotic behavior of integrals 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 (due to Kawamata): 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. In particular, this gives Kawamata’s results 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. We will also provide some introduction to analytic notions involved.