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Applications

  • Antoine Chambert-Loir
  • Johannes Nicaise
  • Julien Sebag
Chapter
Part of the Progress in Mathematics book series (PM, volume 325)

Abstract

This final chapter is devoted to a selection of notable applications of motivic integration.

Bibliography

  1. S. Abhyankar (1956), On the valuations centered in a local domain. Am. J. Math. 78, 321–348MathSciNetzbMATHGoogle Scholar
  2. N. A’Campo (1975), La fonction zêta d’une monodromie. Comment. Math. Helv. 50, 233–248MathSciNetzbMATHGoogle Scholar
  3. F. Ambro (1999), On minimal log discrepancies. Math. Res. Lett. 6(5–6), 573–580MathSciNetzbMATHGoogle Scholar
  4. Y. André (2004), Une introduction aux motifs (motifs purs, motifs mixtes, périodes). Panoramas et Synthèses 17 (Soc. Math. France)Google Scholar
  5. M. Artin (1986), Néron models, in Arithmetic Geometry (Storrs, Connecticut, 1984) (Springer, New York), pp. 213–230Google Scholar
  6. J. Ayoub (2007a/2008), Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique. I. Astérisque 314, x+466 pp.Google Scholar
  7. J. Ayoub (2007b/2008), Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique. II. Astérisque 315, vi+364 pp.Google Scholar
  8. J. Ayoub (2015), Motifs des variétés analytiques rigides. Mém. Soc. Math. Fr. (N.S.) 140–141, vi+386Google Scholar
  9. J. Ayoub, F. Ivorra, J. Sebag (2017), Motives of rigid analytic tubes and nearby motivic sheaves. Ann. Sci. École Norm. Sup. 50(6), 1335–1382MathSciNetzbMATHGoogle Scholar
  10. V.V. Batyrev (1999a), Birational Calabi-Yau n-folds have equal Betti numbers, in New Trends in Algebraic Geometry (Warwick, 1996). London Mathematical Society, Lecture Note Series, vol. 264 (Cambridge University Press, Cambridge), pp. 1–11Google Scholar
  11. V.V. Batyrev (1999b), Non-Archimedean integrals and stringy Euler numbers of log-terminal pairs. J. Eur. Math. Soc. 1(1), 5–33MathSciNetzbMATHGoogle Scholar
  12. V.V. Batyrev, D.I. Dais (1996), Strong McKay correspondence, string-theoretic Hodge numbers and mirror symmetry. Topology 35(4), 901–929MathSciNetzbMATHGoogle Scholar
  13. A. Beauville (1983), Variétés Kähleriennes dont la première classe de Chern est nulle. J. Differ. Geom. 18(4), 755–782zbMATHGoogle Scholar
  14. V.G. Berkovich (1993), Étale cohomology for non-Archimedean analytic spaces. Publ. Math. Inst. Hautes Études Sci. 78, 5–161zbMATHCrossRefGoogle Scholar
  15. V.G. Berkovich (1996a), Vanishing cycles for formal schemes. II. Invent. Math. 125(2), 367–390MathSciNetzbMATHGoogle Scholar
  16. V.G. Berkovich (1996b), Vanishing cycles for non-Archimedean analytic spaces. J. Am. Math. Soc. 9(4), 1187–1209MathSciNetzbMATHGoogle Scholar
  17. S. Bosch, W. Lütkebohmert, M. Raynaud, Néron Models (1990), Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 21 (Springer, Berlin)Google Scholar
  18. S. Bosch, W. Lütkebohmert, M. Raynaud (1995), Formal and rigid geometry. III. The relative maximum principle. Math. Ann. 302(1), 1–29zbMATHGoogle Scholar
  19. E. Bultot, J. Nicaise (2016), Computing motivic zeta functions on log smooth models. arXiv:1610.00742Google Scholar
  20. J. Burillo (1990), El polinomio de Poincaré-Hodge de un producto simétrico de variedades kählerianas compactas. Collect. Math. 41(1), 59–69MathSciNetGoogle Scholar
  21. W. Chen, Y. Ruan (2004), A new cohomology theory of orbifold. Commun. Math. Phys. 248(1), 1–31MathSciNetzbMATHGoogle Scholar
  22. O. Debarre, A. Laface, R. Xavier (2017), Lines on cubic hypersurfaces over finite fields, in Geometry over Nonclosed Fields (Simons Publications, New York). arXiv:1510.05803Google Scholar
  23. T. de Fernex (2013), Three-dimensional counter-examples to the Nash problem. Compos. Math. 149(9), 1519–1534. http://dx.doi.org/10.1112/S0010437X13007252. arXiv:1205.0603
  24. T. de Fernex, R. Docampo (2016), Terminal valuations and the Nash problem. Invent. Math. 203(1), 303–331MathSciNetzbMATHGoogle Scholar
  25. A.J. de Jong (1995/1996), Crystalline Dieudonné module theory via formal and rigid geometry. Inst. Hautes Études Sci. Publ. Math. 82, 5–96MathSciNetzbMATHGoogle Scholar
  26. S. del Baño Rollin, V. Navarro Aznar (1998), On the motive of a quotient variety. Collect. Math. 49(2–3), 203–226. Dedicated to the memory of Fernando SerranoGoogle Scholar
  27. J.-P. Demailly, J. Kollár (2001), Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds. Ann. Sci. École Norm. Sup. (4) 34(4), 525–556MathSciNetzbMATHGoogle Scholar
  28. J. Denef, F. Loeser (1998), Motivic Igusa zeta functions. J. Algebraic Geom. 7(3), 505–537MathSciNetzbMATHGoogle Scholar
  29. J. Denef, F. Loeser (2001), Geometry on arc spaces of algebraic varieties, in European Congress of Mathematics, Volume I (Barcelona, 2000). Progress in Mathematics, vol. 201 (Birkhäuser, Basel), pp. 327–348zbMATHGoogle Scholar
  30. J. Denef, F. Loeser (2002a), Lefschetz numbers of iterates of the monodromy and truncated arcs. Topology 41(5), 1031–1040MathSciNetzbMATHGoogle Scholar
  31. J. Denef, F. Loeser (2002b), Lefschetz numbers of iterates of the monodromy and truncated arcs. Topology 41(5), 1031–1040MathSciNetzbMATHGoogle Scholar
  32. J. Denef, F. Loeser (2002c), Motivic integration, quotient singularities and the McKay correspondence. Compos. Math. 131(3), 267–290MathSciNetzbMATHGoogle Scholar
  33. J. Denef, F. Loeser (2004), On some rational generating series occurring in arithmetic geometry, in Geometric Aspects of Dwork Theory, vols. I, II (Walter de Gruyter, Berlin), pp. 509–526Google Scholar
  34. B. Dwork (1960), On the rationality of the zeta function of an algebraic variety. Am. J. Math. 82, 631–648MathSciNetzbMATHGoogle Scholar
  35. L. Ein, M. Mustaţǎ (2004), Inversion of adjunction for local complete intersection varieties. Am. J. Math. 126(6), 1355–1365MathSciNetzbMATHGoogle Scholar
  36. L. Ein, M. Mustaţă, T. Yasuda (2003), Jet schemes, log discrepancies and inversion of adjunction. Invent. Math. 153(3), 519–535MathSciNetzbMATHGoogle Scholar
  37. L. Ein, R. Lazarsfeld, M. Mustaţǎ (2004), Contact loci in arc spaces. Compos. Math. 140(5), 1229–1244MathSciNetzbMATHCrossRefGoogle Scholar
  38. H. Esnault, J. Nicaise (2011), Finite group actions, rational fixed points and weak Néron models. Pure Appl. Math. Q. 7(4), 1209–1240. Special Issue: In memory of Eckart ViehwegMathSciNetzbMATHGoogle Scholar
  39. J. Fernández de Bobadilla (2012), Nash problem for surface singularities is a topological problem. Adv. Math. 230(1), 131–176MathSciNetzbMATHGoogle Scholar
  40. J. Fernández de Bobadilla, M. Pe Pereira (2012), The Nash problem for surfaces. Ann. Math. (2) 176(3), 2003–2029Google Scholar
  41. J. Fogarty (1968), Algebraic families on an algebraic surface. Am. J. Math. 90, 511–521. http://dx.doi.org/10.2307/2373541 MathSciNetzbMATHGoogle Scholar
  42. S. Galkin, E. Shinder (2014), The Fano variety of lines and rationality problem for a cubic hypersurface. arXiv:1405.5154Google Scholar
  43. L. Göttsche (2001), On the motive of the Hilbert scheme of points on a surface. Math. Res. Lett. 8(5–6), 613–627MathSciNetzbMATHGoogle Scholar
  44. A. Grothendieck (1971), Revêtements étales et groupe fondamental — SGA I. Lecture Notes in Mathematics, vol. 224 (Springer, Berlin). Quoted as ((alias?))Google Scholar
  45. L.H. Halle, J. Nicaise (2011), Motivic zeta functions of abelian varieties, and the monodromy conjecture. Adv. Math. 227(1), 610–653MathSciNetzbMATHGoogle Scholar
  46. L.H. Halle, J. Nicaise (2016), Néron Models and Base Change. Lecture Notes in Mathematics, vol. 2156 (Springer, Cham)zbMATHCrossRefGoogle Scholar
  47. L.H. Halle, J. Nicaise (2017), Motivic zeta functions of degenerating Calabi-Yau varieties. Math. Ann., arXiv:1701.09155Google Scholar
  48. A. Hartmann (2015), Equivariant motivic integration on formal schemes and the motivic zeta function. arXiv:1511.08656Google Scholar
  49. O. Haution (2017), On rational fixed points of finite group actions on the affine space. Trans. Am. Math. Soc. 369(11), 8277–8290. http://dx.doi.org/10.1090/tran/7184 MathSciNetzbMATHGoogle Scholar
  50. F. Heinloth (2007), A note on functional equations for zeta functions with values in Chow motives. Ann. Inst. Fourier (Grenoble) 57(6), 1927–1945MathSciNetzbMATHGoogle Scholar
  51. E. Hrushovski, F. Loeser (2015), Monodromy and the Lefschetz fixed point formula. Ann. Sci. Éc. Norm. Supér. (4) 48(2), 313–349MathSciNetzbMATHGoogle Scholar
  52. L. Illusie (1981), Théorie de Brauer et caractéristique d’Euler-Poincaré (d’après P. Deligne). The Euler-Poincaré characteristic (French), Astérisque 82, pp. 161–172, Soc. Math. France, ParisGoogle Scholar
  53. S. Ishii (2008), Maximal divisorial sets in arc spaces, in Algebraic Geometry in East Asia—Hanoi 2005. Advanced Studies in Pure Mathematics, vol. 50 (Mathematical Society of Japan, Tokyo), pp. 237–249Google Scholar
  54. S. Ishii, J. Kollár (2003), The Nash problem on arc families of singularities. Duke Math. J. 120(3), 601–620MathSciNetzbMATHCrossRefGoogle Scholar
  55. S. Ishii, A.J. Reguera (2013), Singularities with the highest Mather minimal log discrepancy. Math. Z. 275(3–4), 1255–1274MathSciNetzbMATHCrossRefGoogle Scholar
  56. T. Ito (2004), Stringy Hodge numbers and p-adic Hodge theory. Compos. Math. 140(6), 1499–1517MathSciNetzbMATHGoogle Scholar
  57. F. Ivorra (2014), Finite dimension motives and applications (following S-I. Kimura, P. O’Sullivan and others). Autour des motifs, II. Asian-French summer school on algebraic geometry and number theory. Panoramas et synthèses, vol. 38, Soc. Math. FranceGoogle Scholar
  58. F. Ivorra, J. Sebag (2012), Géométrie algébrique par morceaux, K-équivalence et motifs. Enseign. Math. (2), 58, 375–403MathSciNetzbMATHGoogle Scholar
  59. F. Ivorra, J. Sebag (2013), Nearby motives and motivic nearby cycles. Selecta Math. (N.S.) 19(4), 879–902MathSciNetzbMATHGoogle Scholar
  60. J.M. Johnson, J. Kollár (2013), Arc spaces of cA-type singularities. J. Singul. 7, 238–252MathSciNetzbMATHGoogle Scholar
  61. B. Kahn (2009), Zeta functions and motives. Pure Appl. Math. Q. 5(1), 507–570MathSciNetzbMATHGoogle Scholar
  62. M. Kapranov (2000), The elliptic curve in the S-duality theory and Eisenstein series for Kac-Moody groups. arXiv:math/0001005Google Scholar
  63. J. Kollár (ed.) (1992), Flips and abundance for algebraic threefolds, Société Mathématique de France, Paris. Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, August 1991, Astérisque No. 211 (1992) (1992)Google Scholar
  64. J. Kollár (1996), Rational Curves on Algebraic Varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 32 (Springer, Berlin)CrossRefGoogle Scholar
  65. J. Kollár, S. Mori (1998), Birational Geometry of Algebraic Varieties. Cambridge Tracts in Mathematics, vol. 134 (Cambridge University Press, Cambridge). With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese originalGoogle Scholar
  66. M. Kontsevich (1995), Motivic integration. Lecture at Orsay. http://www.lama.univ-savoie.fr/~raibaut/Kontsevich-MotIntNotes.pdf
  67. M. Larsen, V.A. Lunts (2003), Motivic measures and stable birational geometry. Mosc. Math. J. 3(1), 85–95, 259. arXiv:math.AG/0110255Google Scholar
  68. M. Larsen, V.A. Lunts (2004), Rationality criteria for motivic zeta functions. Compos. Math. 140(6), 1537–1560MathSciNetzbMATHGoogle Scholar
  69. G. Laumon (1981), Comparaison de caractéristiques d’Euler-Poincaré en cohomologie l-adique. C. R. Acad. Sci. Paris Sér. I Math. 292(3), 209–212MathSciNetzbMATHGoogle Scholar
  70. M. Lejeune-Jalabert, A.J. Reguera (2012), Exceptional divisors that are not uniruled belong to the image of the Nash map. J. Inst. Math. Jussieu 11(2), 273–287MathSciNetzbMATHGoogle Scholar
  71. D. Litt (2015), Zeta functions of curves with no rational points. Michigan Math. J. 64(2), 383–395. arXiv:1405.7380MathSciNetzbMATHGoogle Scholar
  72. F. Loeser, J. Sebag (2003), Motivic integration on smooth rigid varieties and invariants of degenerations. Duke Math. J. 119(2), 315–344MathSciNetzbMATHGoogle Scholar
  73. I.G. Macdonald (1962), The Poincaré polynomial of a symmetric product. Proc. Camb. Philos. Soc. 58, 563–568zbMATHGoogle Scholar
  74. D. Mumford (1974), Abelian Varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5. Published for the Tata Institute of Fundamental Research, Bombay/Oxford University Press, London)Google Scholar
  75. M. Mustaţă (2001), Jet schemes of locally complete intersection canonical singularities. Invent. Math. 145(3), 397–424. With an appendix by David Eisenbud and Edward FrenkelGoogle Scholar
  76. M. Mustaţă (2002), Singularities of pairs via jet schemes. J. Am. Math. Soc. 15(3), 599–615 [electronic]Google Scholar
  77. C. Nakayama (1998), Nearby cycles for log smooth families. Compos. Math. 112(1), 45–75MathSciNetzbMATHGoogle Scholar
  78. J.F. Nash Jr. (1995/1996), Arc structure of singularities. Duke Math. J. 81(1), 31–38. A celebration of John F. Nash, Jr.Google Scholar
  79. J. Nicaise (2009), A trace formula for rigid varieties, and motivic Weil generating series for formal schemes. Math. Ann. 343(2), 285–349MathSciNetzbMATHGoogle Scholar
  80. J. Nicaise (2011b), A trace formula for varieties over a discretely valued field. J. Reine Angew. Math. 650, 193–238MathSciNetzbMATHGoogle Scholar
  81. J. Nicaise (2013), Geometric criteria for tame ramification. Math. Z. 273(3–4), 839–868MathSciNetzbMATHGoogle Scholar
  82. J. Nicaise, J. Sebag (2007a), Motivic Serre invariants of curves. Manuscripta Math. 123(2), 105–132MathSciNetzbMATHGoogle Scholar
  83. J. Nicaise, J. Sebag (2007b), Motivic Serre invariants, ramification, and the analytic Milnor fiber. Invent. Math. 168(1), 133–173MathSciNetzbMATHGoogle Scholar
  84. J. Nicaise, C. Xu (2016), Poles of maximal order of motivic zeta functions. Duke Math. J. 165(2), 217–243MathSciNetzbMATHGoogle Scholar
  85. A.J. Reguera (2006), A curve selection lemma in spaces of arcs and the image of the Nash map. Compos. Math. 142(1), 119–130MathSciNetzbMATHCrossRefGoogle Scholar
  86. J. Schepers (2006), Stringy E-functions of varieties with A-D-E singularities. Manuscripta Math. 119(2), 129–157MathSciNetzbMATHGoogle Scholar
  87. J. Schepers, W. Veys (2007), Stringy Hodge numbers for a class of isolated singularities and for threefolds. Int. Math. Res. Not. 2007(2), Article ID rnm016, 14Google Scholar
  88. J. Schepers, W. Veys (2009), Stringy E-functions of hypersurfaces and of Brieskorn singularities. Adv. Geom. 9(2), 199–217MathSciNetzbMATHGoogle Scholar
  89. J. Sebag (2010b), Variétés K-équivalentes et géométrie par morceaux. Arch. Math. (Basel) 94(3), 207–217. http://dx.doi.org/10.1007/s00013-009-0095-3 MathSciNetzbMATHGoogle Scholar
  90. A. Smeets (2017), Logarithmic good reduction, monodromy and the rational volume. Algebra & Number Theory 11(1), 213–233MathSciNetzbMATHGoogle Scholar
  91. M. Temkin (2008), Desingularization of quasi-excellent schemes in characteristic zero. Adv. Math. 219(2), 488–522MathSciNetzbMATHGoogle Scholar
  92. M. Temkin (2009), Functorial desingularization over \(\mathfrak{q}\): boundaries and the embedded case. Arxiv:0912.2570Google Scholar
  93. A.N. Varchenko (1982), The complex singularity index does not change along the stratum μ = const. Funktsional. Anal. i Prilozhen. 16(1), 1–12, 96MathSciNetzbMATHGoogle Scholar
  94. W. Veys (2004), Stringy invariants of normal surfaces. J. Algebraic Geom. 13(1), 115–141MathSciNetzbMATHGoogle Scholar
  95. T. Yasuda (2004), Twisted jets, motivic measures and orbifold cohomology. Compos. Math. 140(2), 396–422MathSciNetzbMATHGoogle Scholar
  96. O. Zariski (1939), The reduction of the singularities of an algebraic surface. Ann. Math. (2) 40, 639–689MathSciNetzbMATHGoogle Scholar
  97. Z. Zhu (2013), Log canonical thresholds in positive characteristic. arXiv:1308.5445Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Antoine Chambert-Loir
    • 1
  • Johannes Nicaise
    • 2
  • Julien Sebag
    • 3
  1. 1.Université Paris Diderot, Sorbonne Paris CitéInstitut de Mathématiques de Jussieu-Paris Rive GaucheParisFrance
  2. 2.Department of MathematicsImperial College LondonLondonUK
  3. 3.IrmarUniversité de Rennes 1Rennes CedexFrance

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