Motivic zeta functions of degenerating Calabi–Yau varieties

Halle, Lars Halvard; Nicaise, Johannes

doi:10.1007/s00208-017-1578-3

Cited by 13 publications

(25 citation statements)

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“…The behavior of the motivic volume under finite extensions of K is encoded in Denef and Loeser's motivic Igusa zeta function via its interpretation in [NS07]. If K has equal characteristic zero, the leading asymptotic term of the motivic zeta functions of Calabi-Yau varieties was studied in [HN17], and the relations with the Kontsevich-Soibelman skeleton are highlighted in [HN17, 3.2.3]. These results should be viewed as a geometric version of our convergence statements for non-archimedean local fields.…”

Section: Related Resultsmentioning

confidence: 99%

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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Section: Related Resultsmentioning

confidence: 99%

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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“…We shall also see that there is a close relationship between the dual complexes of the strict Kulikov models of A F and X F . Kulikov models have been applied to the study of motivic zeta functions of K3-surfaces and the monodromy conjecture ( [12], Definition 2.3.5), and our results will provide a new class of K3 surfaces which satisfy the monodromy property.…”

Section: Introductionmentioning

confidence: 93%

“…Throughout this section (except for the final corollary), we shall assume that the residue field k of O K is of characteristic zero. For a precise statement of the monodromy conjecture (or rather a refined version thereof), see [12], Definition 2.3.5. Roughly speaking, it can be summarized as follows: If X is a smooth, projective, geometrically integral algebraic variety over K with trivial canonical bundle (generated by a global top-form which we call ω), we can consider the motivic Zeta function Z X,ω (t), which is an element of the ring M µ k [[t]], where M µ k := K µ 0 (Var k )[L −1 ].…”

Section: Equivariant Kulikov Models Of Kummer Surfaces and The Monodmentioning

confidence: 99%

“…Roughly speaking, it can be summarized as follows: If X is a smooth, projective, geometrically integral algebraic variety over K with trivial canonical bundle (generated by a global top-form which we call ω), we can consider the motivic Zeta function Z X,ω (t), which is an element of the ring M µ k [[t]], where M µ k := K µ 0 (Var k )[L −1 ]. Here, K µ 0 (Var k ) denotes the µ−equivariant Grothendieck ring of varieties; see [12], (2.2.1) and Definition 2.3.1 of loc. cit.…”

Section: Equivariant Kulikov Models Of Kummer Surfaces and The Monodmentioning

confidence: 99%

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Degeneration of Kummer surfaces

Overkamp

2020

Math. Proc. Camb. Phil. Soc.

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We prove that a Kummer surface defined over a complete strictly Henselian discretely valued field K of residue characteristic different from 2 admits a strict Kulikov model after finite base change. The Kulikov models we construct will be schemes, so our results imply that the semistable reduction conjecture is true for Kummer surfaces in this setup, even in the category of schemes. Our construction of Kulikov models is closely related to an earlier construction of Künnemann, which produces semistable models of Abelian varieties. It is well-known that the special fibre of a strict Kulikov model belongs to one of three types, and we shall prove that the type of the special fibre of a strict Kulikov model of a Kummer surface and the toric rank of a corresponding Abelian surface are determined by each other. We also study the relationship between this invariant and the Galois representation on the second ℓ-adic cohomology of the Kummer surface. Finally, we apply our results, together with earlier work of Halle-Nicaise, to give a proof of the monodromy conjecture for Kummer surfaces in equal characteristic zero.

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“…We have already applied our formula to compute the motivic zeta function of the degeneration of the quartic surface in [NOR16]. Our formula is also used in an essential way in [HN16] to prove an analog of the monodromy conjecture for a large and interesting class of degenerations of Calabi-Yau varieties (namely, the degenerations with monodromy-equivariant Kulikov models).…”

Section: Introductionmentioning

confidence: 99%

Computing motivic zeta functions on log smooth models

Bultot

Nicaise

2019

Math. Z.

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We give an explicit formula for the motivic zeta function in terms of a log smooth model. It generalizes the classical formulas for snc-models, but it gives rise to much fewer candidate poles, in general. This formula plays an essential role in recent work on motivic zeta functions of degenerating Calabi–Yau varieties by the second-named author and his collaborators. As a further illustration, we explain how the formula for Newton non-degenerate polynomials can be viewed as a special case of our results.

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Motivic zeta functions of degenerating Calabi–Yau varieties

Cited by 13 publications

References 56 publications

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

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

Degeneration of Kummer surfaces

Computing motivic zeta functions on log smooth models

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