On semipositivity theorems

Fujino, Osamu; Fujisawa, Taro

doi:10.48550/arxiv.1701.02039

Cited by 8 publications

(18 citation statements)

References 11 publications

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“…It is a consequence of deep results in the theory of variation of Hodge structure by Cattani, Kaplan and Schmid [CKS86] (cf. also [Ks85] and some more related references listed in [FF17], [Br17]). Here the asymptotics is given by a plurisubharmonic weight with vanishing Lelong numbers in the special case of the unipotent monodromies condition, which amounts to generalizing the logarithmic singularity log |z| 2 β .…”

Section: Introductionmentioning

confidence: 98%

“…In contrast to the above numerous previous results when the base dimension dim Y is equal to 1, the only previous result to our knowledge in the general case of dim Y ≥ 2 is the asymptotics of the L 2 metric for the direct image f * (K X/Y ) given in the Kawamata semipositivity theorem (see Theorem 5.1) [Ka00] (cf. [Ka81]), [FF17], [Br17] under the unipotent monodromies condition. It is a consequence of deep results in the theory of variation of Hodge structure by Cattani, Kaplan and Schmid [CKS86] (cf.…”

Section: Introductionmentioning

confidence: 99%

“…Next we explain how the individual items in Question 1.6 follow from Theorems 1.3, 1.4. For [Ka98], the nefness of the moduli part line bundle J follows immediately from Theorems 1.3, 1.4 since a Q-line bundle J admitting a singular hermitian metric with vanishing Lelong numbers (we will call this property of J as pseudo-semiample) is nef [D92], [FF17].…”

Section: Introductionmentioning

confidence: 99%

“…For [F78], [Ka81], [Ka00], Theorem 1.3 gives a new proof for the following semipositivity theorem due to [Ka00,Theorem 1.1(3)], [FF17] (cf. [Ko87]) for f * (K X/Y ) when it is a line bundle (which had strengthened its nefness obtained in [F78], [Ka81]).…”

Section: Introductionmentioning

confidence: 99%

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Canonical bundle formula and degenerating families of volume forms

Kim

2019

Preprint

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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.

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Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Canonical bundle formula and degenerating families of volume forms

Kim

2019

Preprint

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“…Then N q = (B −1 ⊗K q )∩F q . By the work of Zuo [Zuo00] (see also [PW16,Bru17,FF17,Bru18] for various generalizations) on the negativity of kernels of Kodaira-Spencer maps of Hodge bundles, K * q is weakly positive in the sense of Viehweg 2 , cf. [VZ02, Lemma 4.4.(v)].…”

Section: Construction Of the Viehweg-zuo Higgs Bundlementioning

confidence: 99%

Hyperbolicity of bases of log Calabi-Yau families

Deng

2019

Preprint

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Algebraic fiber spaces and curvature of higher direct images

Berndtsson,

Paun,

Wang

2017

Preprint

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Let p : X → Y be an algebraic fiber space, and let L be a line bundle on X. In this article we obtain a curvature formula for the higher direct images of Ω i X/Y ⊗ L restricted to a suitable Zariski open subset of X. Our results are particularly meaningful in case L is seminegatively curved on X and strictly negative or trivial on smooth fibers of p. Several applications are obtained, including a new proof of a result by Viehweg-Zuo in the context of canonically polarized family of maximal variation and its version for Calabi-Yau families. The main feature of our approach is that the general curvature formulas we obtain allow us to bypass the use of ramified covers -and the complications which are induced by them.

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On semipositivity theorems

Cited by 8 publications

References 11 publications

Canonical bundle formula and degenerating families of volume forms

Canonical bundle formula and degenerating families of volume forms

Hyperbolicity of bases of log Calabi-Yau families

Algebraic fiber spaces and curvature of higher direct images

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