Generalised pairs in birational geometry

Birkar, Caucher

doi:10.48550/arxiv.2008.01008

Cited by 7 publications

(7 citation statements)

References 43 publications

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“…Generalised pairs, introduced in [12], have found many applications in higher dimensional algebraic geometry in recent years; see [4] for a survey. A projective generalised pair (X, B + M ) consists of a normal projective variety X, an R-divisor B with non-negative coefficients, and an R-divisor M which is the pushdown of a nef R-divisor M ′ on some birational model X ′ → X.…”

Section: Introductionmentioning

confidence: 99%

Boundedness and volume of generalised pairs

Birkar¹

2021

Preprint

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In this paper we investigate boundedness and volumes of generalised pairs, and give applications to usual pairs especially to a class of pairs that we call stable log minimal models.Fixing the dimension and a DCC set controlling coefficients, we will show that the set of volumes of all projective generalised lc pairs (X, B + M ) under the given data, satisfies the DCC. Futhermore, we will show that in the klt case, the set of such pairs with ample KX + B + M and fixed volume, forms a bounded family.We prove a result about descent of nef divisors to bounded families. This is the key to proving the above and various other results.We will then apply the above to study projective lc pairs (X, B) with abundant KX +B of arbitrary Kodaira dimension. In particular, we show that the set of Iitaka volumes of such pairs satisfies DCC under some natural boundedness assumptions on the fibres of the Iitaka fibration.We define stable log minimal models which consist of a projective lc pair (X, B) with semi-ample KX + B together with a divisor A ≥ 0 so that KX + B + A is ample and A does not contain any non-klt centre of (X, B). This is a generalisation of both usual stable pairs of general type and stable log Calabi-Yau pairs. Fixing appropriate invariants we show that stable log minimal models form a bounded family. Then we discuss connection with moduli spaces. Caucher Birkar3.4. Bounded coefficients on generalised lc modifications 18 3.6. Bounded crepant models 21 3.8. Finiteness of generalised lc thresholds on bounded families 24 3.10. Construction of towers of crepant models 25 3.11. Boundedness of length of towers of crepant models 26 3.13. Descent of nef divisors to bounded models 27 4. DCC of volume of generalised pairs 30 4.1. DCC of volumes of pairs with fixed birational model 31 4.3. DCC of volumes for G glc (d, Φ) 34 5. Bounded birational models for G glc (d, Φ, v) 36 6. Boundedness of generalised pairs 39 6.1. Decrease of volume 39 6.4. Log discrepancies of F glc (d, Φ, v) 41 6.6. Boundedness of F gklt (d, Φ, v) 46 7. DCC of Iitaka volumes 48 7.1. Adjunction formula for I lc (d, Φ, u) 48 7.5. DCC of Iitaka volumes 52 8. Boundedness of stable log minimal models 53 8.1. Log discrepancies on stable log minimal models 53 8.4. Boundedness of S lc (d, Φ, u, v, σ) and S klt (d, Φ, u, v,

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Section: Introductionmentioning

confidence: 99%

Boundedness and volume of generalised pairs

Birkar¹

2021

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“…For other results closely related to the theory of generalized pairs, we refer the reader to [HX15, Fil18a, Mor18, HL18, Fil18b, Bir18, HH19, HL19, LT19, HM20, HL20a, HL20b, HL20d, LP20a, LP20b, Li20, Hu20, FS20a, Fil20, Bir20a, HL20c, CX20, Bir20c, FS20b, FW20, BDCS20, CT20, Sho20, Has20, Li21, Liu21, LX21, Hu21, Jia21, Bir21b]. We also refer the reader to [Bir20b] for a more detailed introduction to the theory of generalized pairs. It has recently become apparent that the minimal model program (MMP) for generalized pairs is closely related to the minimal model program for usual pairs and varieties.…”

Section: Introductionmentioning

confidence: 99%

Existence of flips for generalized lc pairs

Hacon¹,

Liu²

2021

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We prove the existence of flips for Q-factorial NQC generalized lc pairs, and the cone and contraction theorems for NQC generalized lc pairs. This answers a question of C. Birkar. As an immediate application, we show that we can run the minimal model program for Q-factorial NQC generalized lc pairs. In particular, we complete the minimal model program for Q-factorial NQC generalized lc pairs in dimension ≤ 3 and pseudo-effective Q-factorial NQC generalized lc pairs in dimension 4.

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“…The concept of a generalized pair (g-pair for short), which was introduced in [BZ16] and which generalizes the notion of a usual pair, underlies many of the latest developments in birational geometry. The majority of these developments are outlined in [Bir20], while for further applications of gpairs we refer to the papers [HLi20, HLiu20, LMT20, Che20, CX20], just to name a few. Not only have g-pairs been implemented successfully in a wide range of contexts, but also the recent papers [Mor18,HM18,HLi18,LT19] indicate that it is in fact essential to understand their birational geometry, even if one is only interested in studying the birational geometry of varieties and investigating the main open problems in the Minimal Model Program (MMP) for usual pairs.…”

Section: Introductionmentioning

confidence: 99%

On the termination of flips for log canonical generalized pairs

Chen¹,

Tsakanikas²

2020

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We prove the termination of flips for 4-dimensional pseudoeffective NQC log canonical generalized pairs. As main ingredients, we verify the termination of flips for 3-dimensional NQC log canonical generalized pairs, and show that the termination of flips for pseudoeffective NQC log canonical generalized pairs which admit NQC weak Zariski decompositions follows from the termination of flips in lower dimensions.2010 Mathematics Subject Classification: 14E30.

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Generalised pairs in birational geometry

Abstract: In this note we introduce generalised pairs from the perspective of the evolution of the notion of space in birational algebraic geometry. We describe some applications of generalised pairs in recent years and then mention a few open problems.

Cited by 7 publications

References 43 publications

Boundedness and volume of generalised pairs

Boundedness and volume of generalised pairs

Existence of flips for generalized lc pairs

On the termination of flips for log canonical generalized pairs

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