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. In our main results, we determine the volume asymptotics (equivalently the asymptotics of L 2 metrics) in all base dimensions, which generalizes 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 (due to Kawamata and others): the L 2 metric carries the singularity equal to the discriminant divisor and the moduli part line bundle has a singular hermitian metric with vanishing Lelong numbers. This solves a problem which is implicit in Kawamata's work and recently raised again by Eriksson, Freixas i Montplet and Mourougane.As consequences, we strengthen the semipositivity theorems 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. Instead, in the proof of our main results, together with the study of plurisubharmonic singularities, we use Berndtsson type results on the positivity of direct images. The fact that this much simpler and direct method can replace the use of Hodge theory in the proof of the semipositivity theorems answers a question of Berndtsson.

$L^2$ extension of holomorphic functions for log canonical pairs

“…1.3] 11 taking g : E → Y ′ in the place of X → Y in that statement. (Note that the fiber integral of [K19,Cor. 1.3] corresponds precisely to the direct image of a measure, cf.…”

“…There is a typo of 2a i in (4),[K19, Cor. 1.3] in the current version which should be corrected to −2a i .…”

On the collapsing of Calabi-Yau manifolds and Kähler-Ricci flows

“…, and the asymptotic behavior of the volume form π * ω n can was obtained in [28, Theorems 2.3 and 7.1] using Hodge theory (and in [40] with a different method, which also extends to the case when the morphism f is Kähler but not projective, see [40,Rmk 1.6]): on Ñ \E we have…”

“…The canonical bundle formula. The exponents β j , γ i in (3.1) can be determined by applying the canonical bundle formula in birational geometry [1,21,24,39,40,42,21] to the map f . Following the notation in [40], we define divisors R = − D on M and M = −K Ñ on Ñ , so that we have the equality as Q-divisors…”

“…To prove this result, we refine the arguments in [28] and identify a key divisor Ξ Ñ on the resolution Ñ with the property that whenever Ξ Ñ is π-ample then the desired estimate (1.4) can be shown to hold. We then describe Ξ Ñ explicitly using the canonical bundle formula and the recent results of Kim [40].…”