On the moduli b-divisors of lc-trivial fibrations

“…Thus, as h is finite, we can apply [19,Theorem 1.20] to −K Y + tA for A ample on Y and 0 < t ≪ 1 to conclude that −K Y is pseudo-effective. Now, using ideas of Fujino and Gongyo [17], we study the relation between the generalized pair induced on Z ′ by (X ′ , B ′ + M ′ ) and the one induced by a generalized log canonical center of (X ′ , B ′ + M ′ ) dominating Z ′ .…”

Section: The Case Of Effective Boundarymentioning

confidence: 99%

On a generalized canonical bundle formula and generalized adjunction

Filipazzi¹

2020

Annali Scuola Normale Superiore - Classe Di Scienze

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In this note, we extend the theories of the canonical bundle formula and adjunction to the case of generalized pairs. As an application, we study a particular case of a conjecture by Prokhorov and Shokurov. 2010 Mathematics Subject Classification. 14N30; 14E30, 14J40.1 Theorem 1.8. Let (X, B) be a sub-pair, with coeff(B) ∈ Q. Let f : X → Z be a projective surjective morphism of normal varieties with connected fibers and dim X − dim Z = 2. Assume K X + B ∼ Q,f 0, and (X, B) is klt but not terminal over the genericAfter reviewing some facts about generalized pairs, we introduce the notion of weak generalized dlt model, which carries analogs to most of the good properties of dlt models [30, cf. Definitions and Notation 1.9]. In Theorem 3.2 we prove that such models exist. Then, we switch the focus to the generalized canonical bundle formula. Once it is established, we apply this machinery to the study of generalized adjunction and inversion of adjunction. We conclude discussing some applications to the conjecture by Prokhorov and Shokurov.Acknowledgements. The author would like to thank his advisor Christopher D. Hacon for suggesting the problem, for his insightful suggestions and encouragement. He benefited from several discussions with Joaquín Moraga and Roberto Svaldi. He would also like to thank Tommaso de Fernex and Karl Schwede for helpful conversations. He is also grateful to Dan Abramovich, Florin Ambro and Kalle Karu for answering his questions.3 He would like to thank Joaquín Moraga for useful remarks on a draft of this work, and Jingjun Han for pointing out the relation between the main result of this work and a theorem of Chen and Zhang. Finally, he would like to express his gratitude to the anonymous referee for the careful report and the many suggestions.Throughout this paper, we will work over an algebraically closed field of characteristic 0. In this section, we review some notions about generalized pairs. To start, we recall the definition of pair and generalized pair.Definition 2.1. A generalized (sub)-pair is the datum of a normal variety X ′ , equipped with projective morphisms X → X ′ → V , where f : X → X ′ is birational and X is normal, an R-(sub)-boundary B ′ , and an R-Cartier divisor M on X which is nef over V and such that K X ′ + B ′ + M ′ is R-Cartier, where M ′ := f * M. We call B ′ the boundary part, and M ′ the nef part.

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“…We refer to [16] for the definitions involved in the notion of lc-trivial fibration. For the purposes of this note, it suffices to notice that the above conditions (i) and (ii) are satisfied if (X, B) is a klt projective pair.…”

Section: The Canonical Bundle Formulamentioning

confidence: 99%

“…Furthermore, the fact that M Z is nef should be thought as a weak analog of the fact that M C = 1 12 j * O P 1 (1) in the case of an elliptic surface. Indeed, thanks to work of Ambro and a subsequent generalization of Fujino and Gongyo [2,16], something more is known about M Z . More precisely, under some technical assumptions, M Z is the pull-back of a nef and big divisor on a variety T .…”

Section: The Canonical Bundle Formulamentioning

confidence: 99%

Invariance of Plurigenera and boundedness for Generalized Pairs

Filipazzi¹,

Svaldi²

2020

RMC

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In this note, we survey some recent developments in birational geometry concerning the boundedness of algebraic varieties. We delineate a strategy to extend some of these results to the case of generalized pairs, first introduced by Birkar and Zhang, when the associated log canonical divisor is ample, and the volume is fixed. In this context, we show a version of deformation invariance of plurigenera for generalized pairs. We conclude by discussing an application to the boundedness of varieties of Kodaira dimension κ(X) = dim(X) − 1.

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On the moduli b-divisors of lc-trivial fibrations

Cited by 41 publications

References 18 publications

Variations of Mixed Hodge Structure and Semipositivity Theorems

Variations of Mixed Hodge Structure and Semipositivity Theorems

On a generalized canonical bundle formula and generalized adjunction

Invariance of Plurigenera and boundedness for Generalized Pairs

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