Publications
47 results found
Nicaise J, Ottem JC, 2022, Tropical degenerations and stable rationality, Duke Mathematical Journal, Vol: 171, Pages: 3023-3075, ISSN: 0012-7094
We use the motivic obstruction to stable rationality introduced by Shinderand the first-named author to establish several new classes of stablyirrational hypersurfaces and complete intersections. In particular, we showthat very general quartic fivefolds and complete intersections of a quadric and a cubic in $\mathbb P^6$ are stably irrational. An important new ingredient is the use of tropical degeneration techniques.
Nicaise J, Potemans N, Veys W, 2022, The dlt motivic zeta function is not well-defined, Michigan Mathematical Journal, ISSN: 0026-2285
In [Xu16], Xu defines the dlt motivic zeta function associated toa regular function f on a smooth variety X over a field of characteristic zero.This is an adaptation of the classical motivic zeta function that was introducedby Denef and Loeser [DL98]. The dlt motivic zeta function is defined on adlt modification via a Denef-Loeser-type formula, replacing classes of stratain the Grothendieck ring of varieties by stringy motives. We provide explicitexamples that show that the dlt motivic zeta function depends on the choice ofdlt modification, contrary to what is claimed in [Xu16], and that it is thereforenot well-defined.
Nicaise J, Ottem JC, 2021, A Refinement of the Motivic Volume, and Specialization of Birational Types, RATIONALITY OF VARIETIES, Vol: 342, Pages: 291-322, ISSN: 0743-1643
- Author Web Link
- Cite
- Citations: 1
Jonsson M, Nicaise J, 2020, Convergence of p-adic pluricanonical measures to Lebesgue measures on skeleta in Berkovich spaces, Journal de l'École polytechnique. Mathématiques, Vol: 7, Pages: 287-336, ISSN: 2429-7100
Nicaise J, Shinder E, 2019, The motivic nearby fiber and degeneration of stable rationality, Inventiones Mathematicae, Vol: 217, Pages: 377-413, ISSN: 0020-9910
We prove that stable rationality specializes in regular families whose fibers are integral and have at most ordinary double points as singularities. Our proof is based on motivic specialization techniques and the criterion of Larsen and Lunts for stable rationality in the Grothendieck ring of varieties.
Nicaise J, Payne S, 2019, A tropical motivic Fubini theorem with applications to Donaldson-Thomas theory, Duke Mathematical Journal, Vol: 168, Pages: 1843-1886, ISSN: 0012-7094
We present a new tool for the calculation of Denef and Loeser's motivicnearby fiber and motivic Milnor fiber: a motivic Fubini theorem for thetropicalization map, based on Hrushovski and Kazhdan's theory of motivicvolumes of semi-algebraic sets. As applications, we prove a conjecture ofDavison and Meinhardt on motivic nearby fibers of weighted homogeneouspolynomials, and give a very short and conceptual new proof of the integralidentity conjecture of Kontsevich and Soibelman, first proved by L\^e QuyThuong. Both of these conjectures emerged in the context of motivicDonaldson-Thomas theory.
Bultot E, Nicaise J, 2019, Computing motivic zeta functions on log smooth models, Mathematische Zeitschrift, Vol: 295, Pages: 427-462, ISSN: 0025-5874
We give an explicit formula for the motivic zeta function in terms of a logsmooth model. It generalizes the classical formulas for snc-models, but itgives rise to much fewer candidate poles, in general. As an illustration, weexplain how the formula for Newton non-degenerate polynomials can be viewed asa special case of our results.
Nicaise J, Xu C, Yu TY, 2019, The non-archimedean SYZ fibration, Compositio Mathematica, Vol: 155, Pages: 953-972, ISSN: 0010-437X
We construct non-archimedean SYZ (Strominger–Yau–Zaslow) fibrations for maximally degenerate Calabi–Yau varieties, and we show that they are affinoid torus fibrations away from a codimension-two subset of the base. This confirms a prediction by Kontsevich and Soibelman. We also give an explicit description of the induced integral affine structure on the base of the SYZ fibration. Our main technical tool is a study of the structure of minimal dlt (divisorially log terminal) models along one-dimensional strata.
Nicaise J, Payne S, Schroeter F, 2018, Tropical refined curve counting via motivic integration, Geometry and Topology, Pages: 3175-3234, ISSN: 1364-0380
We propose a geometric interpretation of Block and G\"ottsche's refinedtropical curve counting invariants in terms of virtual$\chi_{-y}$-specializations of motivic measures of semialgebraic sets inrelative Hilbert schemes. We prove that this interpretation is correct forlinear series of genus 1, and in arbitrary genus after specializing from$\chi_{-y}$ to Euler characteristic.
Halle LH, Nicaise J, 2018, Motivic zeta functions of degenerating Calabi-Yau varieties, Mathematische Annalen, Vol: 370, Pages: 1277-1320, ISSN: 0025-5831
We study motivic zeta functions of degenerating families of Calabi-Yauvarieties. Our main result says that they satisfy an analog of Igusa'smonodromy conjecture if the family has a so-called Galois-equivariant Kulikovmodel; we provide several classes of examples where this condition is verified.We also establish a close relation between the zeta function and the skeletonthat appeared in Kontsevich and Soibelman's non-archimedean interpretation ofthe SYZ conjecture in mirror symmetry.
Nicaise J, 2018, Igusa Zeta Functions and the Non-archimedean SYZ Fibration, ACTA MATHEMATICA VIETNAMICA, Vol: 43, Pages: 31-44, ISSN: 0251-4184
Kollar J, Nicaise J, Xu CY, 2018, Semi-stable extensions over 1-dimensional bases, ACTA MATHEMATICA SINICA-ENGLISH SERIES, Vol: 34, Pages: 103-113, ISSN: 1439-8516
Nicaise J, 2018, Geometric invariants for non-archimedean semialgebraic sets, ALGEBRAIC GEOMETRY: SALT LAKE CITY 2015, PT 2, Vol: 97, Pages: 389-403, ISSN: 2324-707X
Baker M, Nicaise J, 2016, Weight functions on Berkovich curves, Algebra & Number Theory, Vol: 10, Pages: 2053-2079, ISSN: 1937-0652
Let $C$ be a curve over a complete discretely valued field $K$. We givetropical descriptions of the weight function attached to a pluricanonical formon $C$ and the essential skeleton of $C$. We show that the Laplacian of theweight function equals the pluricanonical divisor on Berkovich skeleta, and wedescribe the essential skeleton of $C$ as a combinatorial skeleton of theBerkovich skeleton of the minimal $snc$-model. In particular, if $C$ hassemi-stable reduction, then the essential skeleton coincides with the minimalskeleton. As an intermediate step, we describe the base loci of logarithmicpluricanonical line bundles on minimal $snc$-models.
Nicaise J, Xu C, 2016, The essential skeleton of a degeneration of algebraic varieties, American Journal of Mathematics, Vol: 138, Pages: 1645-1667, ISSN: 1080-6377
In this paper, we explore the connections between the Minimal Model Programand the theory of Berkovich spaces. Let $k$ be a field of characteristic zeroand let $X$ be a smooth and proper $k((t))$-variety with semi-ample canonicaldivisor. We prove that the essential skeleton of $X$ coincides with theskeleton of any minimal $dlt$-model and that it is a strong deformation retractof the Berkovich analytification of $X$. As an application, we show that theessential skeleton of a Calabi-Yau variety over $k((t))$ is a pseudo-manifold.
Kesteloot L, Nicaise J, 2016, The specialization index of a variety over a discretely valued field, Proceedings of the American Mathematical Society, Vol: 145, Pages: 585-599, ISSN: 1088-6826
Let $X$ be a proper variety over a henselian discretely valued field. Animportant obstruction to the existence of a rational point on $X$ is the index,the minimal positive degree of a zero cycle on $X$. This paper introduces a newinvariant, the specialization index, which is a closer approximation of theexistence of a rational point. We provide an explicit formula for thespecialization index in terms of an $snc$-model, and we give examples of curveswhere the index equals one but the specialization index is different from one,and thus explains the absence of a rational point. Our main result states thatthe specialization index of a smooth, proper, geometrically connected$\mathbb{C}((t))$-variety with trivial coherent cohomology is equal to one.
Nicaise J, Xu C, 2016, Poles of maximal order of motivic zeta functions, Duke Mathematical Journal, Vol: 165, Pages: 217-243, ISSN: 0012-7094
We prove a 1999 conjecture of Veys, which says that the opposite of the log-canonical threshold is the only possible pole of maximal order of Denef and Loeser’s motivic zeta function associated with a germ of a regular function on a smooth variety over a field of characteristic 0. We apply similar methods to study the weight function on the Berkovich skeleton associated with a degeneration of Calabi–Yau varieties. Our results suggest that the weight function induces a flow on the nonarchimedean analytification of the degeneration towards the Kontsevich–Soibelman skeleton.
Halle LH, Nicaise J, 2016, Néron models and base change, Publisher: Springer, ISBN: 978-3-319-26637-4
We study various aspects of the behaviour of N\'eron models of semi-abelianvarieties under finite extensions of the base field, with a special emphasis onwildly ramified Jacobians. In Part 1, we analyze the behaviour of the componentgroups of the N\'eron models, and we prove rationality results for a certaingenerating series encoding their orders. In Part 2, we discuss Chai's basechange conductor and Edixhoven's filtration, and their relation to the Artinconductor. All of these results are applied in Part 3 to the study of motiviczeta functions of semi-abelian varieties. Part 4 contains some intriguing openproblems and directions for further research. The main tools in this work arenon-archimedean uniformization and a detailed analysis of the behaviour ofregular models of curves under base change.
Nicaise J, 2016, Berkovich skeleta and birational geometry, Nonarchimedean and Tropical Geometry, Editors: Baker, Payne, Publisher: Springer, Pages: 173-194
We give a survey of joint work with Mircea Musta\c{t}\u{a} and Chenyang Xu onthe connections between the geometry of Berkovich spaces over the field ofLaurent series and the birational geometry of one-parameter degenerations ofsmooth projective varieties. The central objects in our theory are the weightfunction and the essential skeleton of the degeneration. We tried to keep thetext self-contained, so that it can serve as an introduction to Berkovichgeometry for birational geometers.
Nicaise J, Overholser DP, Ruddat H, 2016, Motivic zeta functions of the quartic and its mirror dual, Conference on String Math 2014, Publisher: AMER MATHEMATICAL SOC, Pages: 189-200, ISSN: 2324-707X
- Author Web Link
- Cite
- Citations: 1
Mustaţă M, Nicaise J, 2015, Weight functions on non-Archimedean analytic spaces and the Kontsevich–Soibelman skeleton, Algebraic Geometry, Vol: 2, Pages: 365-404
We associate a weight function to pairs (X, ω) consisting of a smooth and proper varietyX over a complete discretely valued field and a pluricanonical form ω on X.This weight function is a real-valued function on the non-Archimedean analytificationof X. It is piecewise affine on the skeleton of any regular model with strict normalcrossings of X, and strictly ascending as one moves away from the skeleton. We applythese properties to the study of the Kontsevich–Soibelman skeleton of (X, ω), andwe prove that this skeleton is connected when X has geometric genus one and ω isa canonical form on X. This result can be viewed as an analog of the Shokurov–Koll´arconnectedness theorem in birational geometry.
Eriksson D, Halle LH, Nicaise J, 2015, A logarithmic interpretation of Edixhoven's jumps for Jacobians, Advances in Mathematics, Vol: 279, Pages: 532-574, ISSN: 0001-8708
Let A be an abelian variety over a discretely valued field. Edixhoven has defined a filtration on the special fiber of the Néron model of A that measures the behavior of the Néron model under tame base change. We interpret the jumps in this filtration in terms of lattices of logarithmic differential forms in the case where A is the Jacobian of a curve C, and we give a compact explicit formula for the jumps in terms of the combinatorial reduction data of C.
Ducros A, Favre C, Nicaise J, 2015, Introduction to the Volume: Berkovich Spaces and Applications, Publisher: SPRINGER INTERNATIONAL PUBLISHING AG, ISBN: 978-3-319-11028-8
Vannieuwenhoven N, Nicaise J, Vandebril R, et al., 2014, ON GENERIC NONEXISTENCE OF THE SCHMIDT-ECKART-YOUNG DECOMPOSITION FOR COMPLEX TENSORS, SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS, Vol: 35, Pages: 886-903, ISSN: 0895-4798
- Author Web Link
- Cite
- Citations: 15
Nicaise J, 2013, Geometric criteria for tame ramification, MATHEMATISCHE ZEITSCHRIFT, Vol: 273, Pages: 839-868, ISSN: 0025-5874
- Author Web Link
- Cite
- Citations: 12
Cluckers R, Loeser F, Nicaise J, 2013, Chai's conjecture and Fubini properties of dimensional motivic integration, ALGEBRA & NUMBER THEORY, Vol: 7, Pages: 893-915, ISSN: 1937-0652
- Author Web Link
- Cite
- Citations: 2
Halle LH, Nicaise J, 2012, Motivic zeta functions for degenerations of abelian varieties and Calabi-Yau varieties, Zeta functions in algebra and geometry, Editors: Campillo, Cardona, Melle-Hernandez, Veys, Zuniga-Galindo, Publisher: American Mathematical Society, Pages: 233-259
This is a survey on motivic zeta functions associated to abelian varietiesand Calabi-Yau varieties over a discretely valued field. We explain how theyare related to Denef and Loeser's motivic zeta function associated to a complexhypersurface singularity and we investigate the relation between the poles ofthe zeta function and the eigenvalues of the monodromy action on the tame$\ell$-adic cohomology of the variety. The motivic zeta function allows togeneralize many interesting arithmetic invariants from abelian varieties toCalabi-Yau varieties and to compute them explicitly on a model with strictnormal crossings.
Halle LH, Nicaise J, 2011, Jumps and monodromy of abelian varieties, Documenta Mathematica, Pages: 937-968, ISSN: 1431-0643
We prove a strong form of the motivic monodromy conjecture for abelianvarieties, by showing that the order of the unique pole of the motivic zetafunction is equal to the size of the maximal Jordan block of the correspondingmonodromy eigenvalue. Moreover, we give a Hodge-theoretic interpretation of thefundamental invariants appearing in the proof.
Cluckers R, Nicaise J, Sebag J, 2011, Motivic Integration and its Interactions with Model Theory and Non-Archimedean Geometry:, Publisher: Cambridge University Press, ISBN: 9780521149761
Applications in several areas are included so that readers can see motivic integration at work in other domains. In a rapidly-evolving area of research this book will prove invaluable.
Halle LH, Nicaise J, 2011, Motivic zeta functions of abelian varieties, and the monodromy conjecture, ADVANCES IN MATHEMATICS, Vol: 227, Pages: 610-653, ISSN: 0001-8708
- Author Web Link
- Cite
- Citations: 23
This data is extracted from the Web of Science and reproduced under a licence from Thomson Reuters. You may not copy or re-distribute this data in whole or in part without the written consent of the Science business of Thomson Reuters.