Log smoothness and polystability over valuation rings

Adiprasito, Karim; Liu, Gaku; Pak, Igor; Temkin, Michael

doi:10.48550/arxiv.1806.09168

Cited by 3 publications

(5 citation statements)

References 7 publications

(12 reference statements)

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“…We begin proving Theorem 1.3 by showing a variant (given in Theorem 6.2) for the pullback to a strictly polystable alteration of X where the support is contained in the union of the canonical faces of the skeleton which are non-degenerate with respect to the alteration. The existence of such a strictly polystable alteration follows from a result of Adiprasito, Liu, Pak and Temkin [ALPT19]. In Theorem 7.7, we will see that the induced morphism from the union of these non-degenerate faces to S X is a piecewise (Q, Γ)-linear surjective map which is finite-to-one.…”

Section: Sincementioning

confidence: 81%

Monge-Ampère measures for toric metrics on abelian varieties

Gubler¹,

Stadlöder²

2022

Preprint

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Toric metrics on a line bundle of an abelian variety A are the invariant metrics under the natural torus action coming from Raynaud's uniformization theory. We compute here the associated Monge-Ampère measures for the restriction to any closed subvariety of A. This generalizes the computation of canonical measures done by the first author from canonical metrics to toric metrics and from discrete valuations to arbitrary nonarchimedean fields. Contents 1. Introduction 1 2. Notation and preliminaries 4 3. Piecewise linear approximation 6 4. Toric metrics 11 5. Strictly polystable alterations 15 6. Monge-Ampère measures of toric metrics 18 7. The canonical subset 19 Appendix A. Differential forms, currents and positivity 24 References 27

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Section: Sincementioning

confidence: 81%

Monge-Ampère measures for toric metrics on abelian varieties

Gubler¹,

Stadlöder²

2022

Preprint

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“…Also the functor j 1 7 does by its explicit description (Theorem 2.9(2)). We deduce then that An ˚pj 1 7 pΛ Y 1 pY 1 qpdqqr2dsq is ι ˚M for some Artin motive M over Y 1an . We let α be the immersion Y Ñ Y 1an .…”

Section: ˙mentioning

confidence: 85%

“…Let S be Spa A. These functors coincide with those induced by the map of analytic varieties f an according to Theorem 2.9 (1). We can therefore use the notation pf an ˚, f an ˚q unambiguously.…”

Section: 2mentioning

confidence: 99%

“…one can require that the alteration e : Y Ñ X (following the notation in loc.cit. ) has degree d which is coprime to ℓ on the dense open sets where it is finite, using [1,Theorem 5.2.18] in place of the alteration proved in [7] and using Lemma A.4. We remark that if e 1 : Y Ñ X is a finite morphism of analytic varieties of degree d coprime to ℓ, then the motive Λ tr pXq is a direct summand of Λ tr pY q since e 1 ˝e1 tr " d ¨Id and d P Λ ˚, where e 1 tr is the transpose of e 1 lying in CorpX, Y q.…”

Section: Appendix a Rigidity For Rigid Motives With Transfers Over Kmentioning

confidence: 99%

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Rigidity for Rigid Analytic Motives

Bambozzi¹,

Vezzani²

2019

J. Inst. Math. Jussieu

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fields, introduced in [5] by Ayoub, both in their version with and without transfers. More precisely, given any normal rigid analytic variety S over K, we denote by RigDAé t pS, Λq (resp. RigDMé t pS, Λq) the category ofétale motives without transfers (resp. with transfers) over S with coefficients in the ring Λ. The precise definition of these categories is recalled in the first section of the paper. Our main result is the following theorem.Theorem (2.1). Let S be a normal rigid analytic variety over a non-Archimedean field K with ℓ-finite cohomological dimension, for all primes ℓ invertible in the residue field k of K, and let Λ be a N-torsion ring, where N is a positive integer invertible in k. The functors:As in the algebraic situation, DpSé t , Λq denotes the derived category of unbounded complexes ofétale sheaves of Λ-modules over the smallétale site and the functors Lι˚arise naturally from the inclusion of the smallétale topos into the big one.We remark that the theorem above is a generalization of the usual Rigidity Theorem, corresponding to the case in which K is trivially valued. Nonetheless, to our knowledge the original algebraic proofs can not be adapted easily to the non-Archimedean context. Our strategy is rather to use algebraic Rigidity to deduce the rigid one, by means of the analytification functors and the relation between rigid varieties and formal schemes. We also remark that, even for proving our statement over a field S " Spa K for motives without transfers, the full relative Rigidity Theorem for schemes is used. Indeed, the six functors formalism plays a crucial role in our proof (see Section 2.2). This is no longer true for motives with transfers, as we show in the appendix, for which a more direct and geometric proof is possible in the absolute case.Just like its algebraic versions, the theorem above has some interesting immediate consequences, discussed in the last section of the paper. They constitute our main motivation for proving the Rigidity Theorem in the non-Archimedean setting.

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“…Any log smooth vertical morphism over O K is log étale locally polystable if the group of values of the valuation ring O K is divisible (e.g. when K is algebraically closed) by [ALPT18,Theorem 5.2.16]. This is also true for the coarser topology divét (see [BPØ22, Definition 3.1.5]).…”

Section: Log Motives and Rigid Motivesmentioning

confidence: 99%

On the p-adic weight-monodromy conjecture for complete intersetions in toric varieties

Binda¹,

Kato²,

Vezzani³

2022

Preprint

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We give a proof of the p-adic weight monodromy conjecture for scheme-theoretic complete intersections in projective smooth toric varieties. The strategy is based on Scholze's proof in the ℓ-adic setting, which we adapt using homotopical results developed in the context of rigid analytic motives CONTENTS 1. Introduction 1 Acknowledgments 5 2. Log motives and rigid motives 5 3. The motivic Hyodo-Kato realization 15 4. Motivic existence of tubular neighborhoods 24 5. The p-adic weight-monodromy for complete intersections 25 Appendix A. A log-free Hyodo-Kato cohomology 28 References 32

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Log smoothness and polystability over valuation rings

Cited by 3 publications

References 7 publications

Monge-Ampère measures for toric metrics on abelian varieties

Monge-Ampère measures for toric metrics on abelian varieties

Rigidity for Rigid Analytic Motives

On the p-adic weight-monodromy conjecture for complete intersetions in toric varieties

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