Computing motivic zeta functions on log smooth models

Bultot, Emmanuel; Nicaise, Johannes

doi:10.48550/arxiv.1610.00742

Cited by 3 publications

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“…This avoids tedious calculations in local coordinates (especially when dealing with the covers E o J in the formula for the motivic volume). The main technical results we will need are already present in [BN16].…”

Section: Appendix a Existence Of The Motivic Volume And Motivic Reduc...mentioning

confidence: 99%

“…For a brief introduction to logarithmic geometry, we refer to [Ka89]. In combination with the overview of regular log schemes in Section 3 of [BN16], this provides all the necessary background material for the arguments presented below. All log structures in this paper are defined with respect to the Zariski topology, as in [Ka94].…”

Section: Appendix a Existence Of The Motivic Volume And Motivic Reduc...mentioning

confidence: 99%

“…All log structures in this paper are defined with respect to the Zariski topology, as in [Ka94]. Following the conventions in [BN16], we will speak of regular log schemes and smooth morphisms of log schemes instead of log regular log schemes and log smooth morphisms. Log schemes will be denoted by symbols of the form X † ; the underlying scheme of X † is denoted by X .…”

Section: Appendix a Existence Of The Motivic Volume And Motivic Reduc...mentioning

confidence: 99%

“…Let ρ be the largest positive integer such that t is divisible by ρ in C X † ,σ . This invariant is called the root index of X † at σ in [BN16]. If the underlying scheme X of X † is regular, then ρ is the greatest common divisor of the multiplicities of the components of X k that pass through σ, by Example 4.1.1 in [BN16].…”

Section: Appendix a Existence Of The Motivic Volume And Motivic Reduc...mentioning

confidence: 99%

“…By [BN16,3.7.1], X (m) is the underlying scheme of the fiber product X (m) † = X † × fs S † S(m) † in the category of fine and saturated log schemes. Since Z † → X † is strict, the fiber product W † = X (m) † × X † Z † in the category of log schemes is already fine and saturated, since it is a strict closed logarithmic subscheme of X (m) † .…”

Section: Appendix a Existence Of The Motivic Volume And Motivic Reduc...mentioning

confidence: 99%

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The motivic nearby fiber and degeneration of stable rationality

Nicaise

Shinder

2019

Invent. math.

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We use the motivic obstruction to stable rationality introduced by Shinder and the first-named author to establish several new classes of stably irrational hypersurfaces and complete intersections. An important new ingredient is the use of tropical degeneration techniques.Notations. We denote by k an algebraically closed field of characteristic zero, and by K the field of Puiseux series over k, that is, ].Moreover, there exists a unique ring homomorphismsuch that the following square commutes:Thus, for every semi-stable proper R-scheme X withProof. The ring homomorphism Vol : K(Var K ) → K(Var k ) from Corollary 3.1.4 in [NS19] sends L to L, and therefore uniquely descends to a ring homomorphism Vol sb : Z[SB K ] → Z[SB k ] by Theorem 2.1.1. The uniqueness of Vol also follows from Corollary 3.1.4 in [NS19]. Thus it suffices to prove that Vol satisfies the equation (2.2.2). Example 2.2.5. Let X be a smooth and proper K-scheme, and assume that X has a smooth and proper R-model X . Then Vol sb ([X] sb ) = [X k ] sb . In particular, Vol sb ([Spec K] sb ) = [Spec k] sb . Corollary 2.2.6. (1) Let X be a smooth and proper K-scheme. If Vol sb ([X] sb ) = [Spec k] sb in Z[SB k ], then X is not stably rational.

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Section: Appendix a Existence Of The Motivic Volume And Motivic Reduc...mentioning

confidence: 99%

Section: Appendix a Existence Of The Motivic Volume And Motivic Reduc...mentioning

confidence: 99%

Section: Appendix a Existence Of The Motivic Volume And Motivic Reduc...mentioning

confidence: 99%

Section: Appendix a Existence Of The Motivic Volume And Motivic Reduc...mentioning

confidence: 99%

Section: Appendix a Existence Of The Motivic Volume And Motivic Reduc...mentioning

confidence: 99%

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The motivic nearby fiber and degeneration of stable rationality

Nicaise

Shinder

2019

Invent. math.

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Motivic zeta functions of degenerating Calabi–Yau varieties

Halle

Nicaise

2017

Math. Ann.

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Abstract. We study motivic zeta functions of degenerating families of Calabi-Yau varieties. Our main result says that they satisfy an analog of Igusa's monodromy conjecture if the family has a so-called Galois equivariant Kulikov model; we provide several classes of examples where this condition is verified. We also establish a close relation between the zeta function and the skeleton that appeared in Kontsevich and Soibelman's non-archimedean interpretation of the SYZ conjecture in mirror symmetry.

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Convergence of p-adic pluricanonical measures to Lebesgue measures on skeleta in Berkovich spaces

Jönsson¹,

Nicaise²

2020

Journal De L’École Polytechnique — Mathématiques

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Let K be a non-archimedean local field, X a smooth and proper K-scheme, and fix a pluricanonical form on X. For every finite extension K of K, the pluricanonical form induces a measure on the K -analytic manifold X(K ). We prove that, when K runs through all finite tame extensions of K, suitable normalizations of the pushforwards of these measures to the Berkovich analytification of X converge to a Lebesgue-type measure on the temperate part of the Kontsevich-Soibelman skeleton, assuming the existence of a strict normal crossings model for X. We also prove a similar result for all finite extensions K under the assumption that X has a log smooth model. This is a non-archimedean counterpart of analogous results for volume forms on degenerating complex Calabi-Yau manifolds by Boucksom and the first-named author. Along the way, we develop a general theory of Lebesgue measures on Berkovich skeleta over discretely valued fields. m K := (π K ) * |θ ⊗ K K | of the measure |θ ⊗ K K | to X an as K runs through finite extensions of K.Fix an algebraic closure K a of K, and let E a K be the set of finite extensions K of K in K a , ordered by inclusion. Let E ur K be the subset of E a K consisting of unramified extensions, and let K ur be the union of all extensions in E ur K (that is, the maximal unramified extension of K in K a ).Main Theorem. Assume X admits a log smooth model over the valuation ring R in K. Then there exist Lebesgue-type measures λ a and λ ur on X an , and positive constants c a K and c ur K for K in E a K and E ur K , respectively, such that lim K ∈E a K c a K m K = λ a , and lim K ∈E ur K c ur K m K = λ ur in the weak sense of positive Radon measures on X an .

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Computing motivic zeta functions on log smooth models

Cited by 3 publications

References 0 publications

The motivic nearby fiber and degeneration of stable rationality

The motivic nearby fiber and degeneration of stable rationality

Motivic zeta functions of degenerating Calabi–Yau varieties

Convergence of p-adic pluricanonical measures to Lebesgue measures on skeleta in Berkovich spaces

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