Anti-pluricanonical systems on Fano varieties

Birkar, Caucher

doi:10.4007/annals.2019.190.2.1

Cited by 132 publications

(257 citation statements)

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“…This proposition allows us to leverage the recent solution of the Borisov-Alexeev-Borisov Conjecture in [Bir16a], [Bir16b]. In these papers Birkar proves that all Fano varieties with mild singularities lie in a bounded family, and in particular, admit a universal upper bound for the anticanonical volume.…”

Section: Varieties With Large A-invariant the Fujita Invariant Was Imentioning

confidence: 87%

On exceptional sets in Manin’s conjecture

Lehmann

Tanimoto

2018

Res Math Sci

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Section: Varieties With Large A-invariant the Fujita Invariant Was Imentioning

confidence: 87%

On exceptional sets in Manin’s conjecture

Lehmann

Tanimoto

2018

Res Math Sci

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“…(I) Boundedness: There is a positive integer N = N(n, V ) such that if X ∈ M Kss n,V (k), then −NK X is a very ample Cartier divisor. This is settled in [Jia17] using results in [Bir16]. (II) Zariski openness: If X → S is a family of Q-Fano varieties, then the locus where the fiber is K-semistable is a Zariski open set.…”

mentioning

confidence: 99%

Uniqueness of K-polystable degenerations of Fano varieties

Blum

2019

Ann. of Math. (2)

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We prove that K-polystable degenerations of Q-Fano varieties are unique. Furthermore, we show that the moduli stack of K-stable Q-Fano varieties is separated. Together with recently proven boundedness and openness statements, the latter result yields a separated Deligne-Mumford stack parametrizing all uniformly K-stable Q-Fano varieties of fixed dimension and volume. The result also implies that the automorphism group of a K-stable Q-Fano variety is finite.

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“…Recently, Birkar and Zhang introduced the notion of generalized pair [8]. This kind of pair arises naturally in certain situations, such as the canonical bundle formula [27,3,18], and adjunction theory [27,3,5]. Furthermore, generalized pairs play an important role in recent developments, such as the study of the Iitaka fibration [8], and the proof of the BAB conjecture [5,7].…”

Section: Introductionmentioning

confidence: 99%

“…In the setup of generalized pairs, Birkar has a version of divisorial inversion of adjunction under some technical conditions. Theorem 1.3 (Lemma 3.2, [5]). Let (X ′ , B ′ + M ′ ) be a Q-factorial generalized pair with data X → X ′ and M. Assume S ′ is a component of B ′ with coefficient 1, and that (X ′ , S ′ ) is plt.…”

Section: Introductionmentioning

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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Anti-pluricanonical systems on Fano varieties

Cited by 132 publications

References 49 publications

On exceptional sets in Manin’s conjecture

On exceptional sets in Manin’s conjecture

Uniqueness of K-polystable degenerations of Fano varieties

On a generalized canonical bundle formula and generalized adjunction

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