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2018 | OriginalPaper | Buchkapitel

7. Applications

verfasst von : Antoine Chambert-Loir, Johannes Nicaise, Julien Sebag

Erschienen in: Motivic Integration

Verlag: Springer New York

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Abstract

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

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Fußnoten
1
Some authors define the center to be the irreducible closed subset of which cX(ν) is the generic point.
 
2
The paper (Ein et al. 2004) calls them simply thin or fat, we explained in remark 4/4.3.7 the change of terminology.
 
3
This paper had been written in 1968.
 
Literatur
Zurück zum Zitat S. Abhyankar (1956), On the valuations centered in a local domain. Am. J. Math. 78, 321–348MathSciNetMATH S. Abhyankar (1956), On the valuations centered in a local domain. Am. J. Math. 78, 321–348MathSciNetMATH
Zurück zum Zitat N. A’Campo (1975), La fonction zêta d’une monodromie. Comment. Math. Helv. 50, 233–248MathSciNetMATH N. A’Campo (1975), La fonction zêta d’une monodromie. Comment. Math. Helv. 50, 233–248MathSciNetMATH
Zurück zum Zitat Y. André (2004), Une introduction aux motifs (motifs purs, motifs mixtes, périodes). Panoramas et Synthèses 17 (Soc. Math. France) Y. André (2004), Une introduction aux motifs (motifs purs, motifs mixtes, périodes). Panoramas et Synthèses 17 (Soc. Math. France)
Zurück zum Zitat M. Artin (1986), Néron models, in Arithmetic Geometry (Storrs, Connecticut, 1984) (Springer, New York), pp. 213–230 M. Artin (1986), Néron models, in Arithmetic Geometry (Storrs, Connecticut, 1984) (Springer, New York), pp. 213–230
Zurück zum Zitat 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. 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.
Zurück zum Zitat 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. 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.
Zurück zum Zitat J. Ayoub (2015), Motifs des variétés analytiques rigides. Mém. Soc. Math. Fr. (N.S.) 140–141, vi+386 J. Ayoub (2015), Motifs des variétés analytiques rigides. Mém. Soc. Math. Fr. (N.S.) 140–141, vi+386
Zurück zum Zitat J. Ayoub, F. Ivorra, J. Sebag (2017), Motives of rigid analytic tubes and nearby motivic sheaves. Ann. Sci. École Norm. Sup. 50(6), 1335–1382MathSciNetMATH J. Ayoub, F. Ivorra, J. Sebag (2017), Motives of rigid analytic tubes and nearby motivic sheaves. Ann. Sci. École Norm. Sup. 50(6), 1335–1382MathSciNetMATH
Zurück zum Zitat 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. 1–11 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. 1–11
Zurück zum Zitat V.V. Batyrev (1999b), Non-Archimedean integrals and stringy Euler numbers of log-terminal pairs. J. Eur. Math. Soc. 1(1), 5–33MathSciNetMATH V.V. Batyrev (1999b), Non-Archimedean integrals and stringy Euler numbers of log-terminal pairs. J. Eur. Math. Soc. 1(1), 5–33MathSciNetMATH
Zurück zum Zitat V.V. Batyrev, D.I. Dais (1996), Strong McKay correspondence, string-theoretic Hodge numbers and mirror symmetry. Topology 35(4), 901–929MathSciNetMATH V.V. Batyrev, D.I. Dais (1996), Strong McKay correspondence, string-theoretic Hodge numbers and mirror symmetry. Topology 35(4), 901–929MathSciNetMATH
Zurück zum Zitat A. Beauville (1983), Variétés Kähleriennes dont la première classe de Chern est nulle. J. Differ. Geom. 18(4), 755–782MATH A. Beauville (1983), Variétés Kähleriennes dont la première classe de Chern est nulle. J. Differ. Geom. 18(4), 755–782MATH
Zurück zum Zitat V.G. Berkovich (1993), Étale cohomology for non-Archimedean analytic spaces. Publ. Math. Inst. Hautes Études Sci. 78, 5–161MATHCrossRef V.G. Berkovich (1993), Étale cohomology for non-Archimedean analytic spaces. Publ. Math. Inst. Hautes Études Sci. 78, 5–161MATHCrossRef
Zurück zum Zitat V.G. Berkovich (1996a), Vanishing cycles for formal schemes. II. Invent. Math. 125(2), 367–390MathSciNetMATH V.G. Berkovich (1996a), Vanishing cycles for formal schemes. II. Invent. Math. 125(2), 367–390MathSciNetMATH
Zurück zum Zitat V.G. Berkovich (1996b), Vanishing cycles for non-Archimedean analytic spaces. J. Am. Math. Soc. 9(4), 1187–1209MathSciNetMATH V.G. Berkovich (1996b), Vanishing cycles for non-Archimedean analytic spaces. J. Am. Math. Soc. 9(4), 1187–1209MathSciNetMATH
Zurück zum Zitat S. Bosch, W. Lütkebohmert, M. Raynaud, Néron Models (1990), Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 21 (Springer, Berlin) S. Bosch, W. Lütkebohmert, M. Raynaud, Néron Models (1990), Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 21 (Springer, Berlin)
Zurück zum Zitat S. Bosch, W. Lütkebohmert, M. Raynaud (1995), Formal and rigid geometry. III. The relative maximum principle. Math. Ann. 302(1), 1–29MATH S. Bosch, W. Lütkebohmert, M. Raynaud (1995), Formal and rigid geometry. III. The relative maximum principle. Math. Ann. 302(1), 1–29MATH
Zurück zum Zitat E. Bultot, J. Nicaise (2016), Computing motivic zeta functions on log smooth models. arXiv:1610.00742 E. Bultot, J. Nicaise (2016), Computing motivic zeta functions on log smooth models. arXiv:1610.00742
Zurück zum Zitat J. Burillo (1990), El polinomio de Poincaré-Hodge de un producto simétrico de variedades kählerianas compactas. Collect. Math. 41(1), 59–69MathSciNet J. Burillo (1990), El polinomio de Poincaré-Hodge de un producto simétrico de variedades kählerianas compactas. Collect. Math. 41(1), 59–69MathSciNet
Zurück zum Zitat W. Chen, Y. Ruan (2004), A new cohomology theory of orbifold. Commun. Math. Phys. 248(1), 1–31MathSciNetMATH W. Chen, Y. Ruan (2004), A new cohomology theory of orbifold. Commun. Math. Phys. 248(1), 1–31MathSciNetMATH
Zurück zum Zitat O. Debarre, A. Laface, R. Xavier (2017), Lines on cubic hypersurfaces over finite fields, in Geometry over Nonclosed Fields (Simons Publications, New York). arXiv:1510.05803MATH O. Debarre, A. Laface, R. Xavier (2017), Lines on cubic hypersurfaces over finite fields, in Geometry over Nonclosed Fields (Simons Publications, New York). arXiv:1510.05803MATH
Zurück zum Zitat T. de Fernex, R. Docampo (2016), Terminal valuations and the Nash problem. Invent. Math. 203(1), 303–331MathSciNetMATH T. de Fernex, R. Docampo (2016), Terminal valuations and the Nash problem. Invent. Math. 203(1), 303–331MathSciNetMATH
Zurück zum Zitat A.J. de Jong (1995/1996), Crystalline Dieudonné module theory via formal and rigid geometry. Inst. Hautes Études Sci. Publ. Math. 82, 5–96MathSciNetMATH A.J. de Jong (1995/1996), Crystalline Dieudonné module theory via formal and rigid geometry. Inst. Hautes Études Sci. Publ. Math. 82, 5–96MathSciNetMATH
Zurück zum Zitat 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 Serrano 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 Serrano
Zurück zum Zitat 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), 525–556MathSciNetMATH 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), 525–556MathSciNetMATH
Zurück zum Zitat 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. 327–348MATH 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. 327–348MATH
Zurück zum Zitat J. Denef, F. Loeser (2002a), Lefschetz numbers of iterates of the monodromy and truncated arcs. Topology 41(5), 1031–1040MathSciNetMATH J. Denef, F. Loeser (2002a), Lefschetz numbers of iterates of the monodromy and truncated arcs. Topology 41(5), 1031–1040MathSciNetMATH
Zurück zum Zitat J. Denef, F. Loeser (2002b), Lefschetz numbers of iterates of the monodromy and truncated arcs. Topology 41(5), 1031–1040MathSciNetMATH J. Denef, F. Loeser (2002b), Lefschetz numbers of iterates of the monodromy and truncated arcs. Topology 41(5), 1031–1040MathSciNetMATH
Zurück zum Zitat J. Denef, F. Loeser (2002c), Motivic integration, quotient singularities and the McKay correspondence. Compos. Math. 131(3), 267–290MathSciNetMATH J. Denef, F. Loeser (2002c), Motivic integration, quotient singularities and the McKay correspondence. Compos. Math. 131(3), 267–290MathSciNetMATH
Zurück zum Zitat 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. 509–526 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. 509–526
Zurück zum Zitat B. Dwork (1960), On the rationality of the zeta function of an algebraic variety. Am. J. Math. 82, 631–648MathSciNetMATH B. Dwork (1960), On the rationality of the zeta function of an algebraic variety. Am. J. Math. 82, 631–648MathSciNetMATH
Zurück zum Zitat L. Ein, M. Mustaţǎ (2004), Inversion of adjunction for local complete intersection varieties. Am. J. Math. 126(6), 1355–1365MathSciNetMATH L. Ein, M. Mustaţǎ (2004), Inversion of adjunction for local complete intersection varieties. Am. J. Math. 126(6), 1355–1365MathSciNetMATH
Zurück zum Zitat L. Ein, M. Mustaţă, T. Yasuda (2003), Jet schemes, log discrepancies and inversion of adjunction. Invent. Math. 153(3), 519–535MathSciNetMATH L. Ein, M. Mustaţă, T. Yasuda (2003), Jet schemes, log discrepancies and inversion of adjunction. Invent. Math. 153(3), 519–535MathSciNetMATH
Zurück zum Zitat 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 ViehwegMathSciNetMATH 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 ViehwegMathSciNetMATH
Zurück zum Zitat J. Fernández de Bobadilla (2012), Nash problem for surface singularities is a topological problem. Adv. Math. 230(1), 131–176MathSciNetMATH J. Fernández de Bobadilla (2012), Nash problem for surface singularities is a topological problem. Adv. Math. 230(1), 131–176MathSciNetMATH
Zurück zum Zitat J. Fernández de Bobadilla, M. Pe Pereira (2012), The Nash problem for surfaces. Ann. Math. (2) 176(3), 2003–2029 J. Fernández de Bobadilla, M. Pe Pereira (2012), The Nash problem for surfaces. Ann. Math. (2) 176(3), 2003–2029
Zurück zum Zitat S. Galkin, E. Shinder (2014), The Fano variety of lines and rationality problem for a cubic hypersurface. arXiv:1405.5154 S. Galkin, E. Shinder (2014), The Fano variety of lines and rationality problem for a cubic hypersurface. arXiv:1405.5154
Zurück zum Zitat L. Göttsche (2001), On the motive of the Hilbert scheme of points on a surface. Math. Res. Lett. 8(5–6), 613–627MathSciNetMATH L. Göttsche (2001), On the motive of the Hilbert scheme of points on a surface. Math. Res. Lett. 8(5–6), 613–627MathSciNetMATH
Zurück zum Zitat A. Grothendieck (1971), Revêtements étales et groupe fondamental — SGA I. Lecture Notes in Mathematics, vol. 224 (Springer, Berlin). Quoted as ((alias?)) A. Grothendieck (1971), Revêtements étales et groupe fondamental — SGA I. Lecture Notes in Mathematics, vol. 224 (Springer, Berlin). Quoted as ((alias?))
Zurück zum Zitat L.H. Halle, J. Nicaise (2011), Motivic zeta functions of abelian varieties, and the monodromy conjecture. Adv. Math. 227(1), 610–653MathSciNetMATH L.H. Halle, J. Nicaise (2011), Motivic zeta functions of abelian varieties, and the monodromy conjecture. Adv. Math. 227(1), 610–653MathSciNetMATH
Zurück zum Zitat L.H. Halle, J. Nicaise (2016), Néron Models and Base Change. Lecture Notes in Mathematics, vol. 2156 (Springer, Cham)MATHCrossRef L.H. Halle, J. Nicaise (2016), Néron Models and Base Change. Lecture Notes in Mathematics, vol. 2156 (Springer, Cham)MATHCrossRef
Zurück zum Zitat L.H. Halle, J. Nicaise (2017), Motivic zeta functions of degenerating Calabi-Yau varieties. Math. Ann., arXiv:1701.09155 L.H. Halle, J. Nicaise (2017), Motivic zeta functions of degenerating Calabi-Yau varieties. Math. Ann., arXiv:1701.09155
Zurück zum Zitat A. Hartmann (2015), Equivariant motivic integration on formal schemes and the motivic zeta function. arXiv:1511.08656 A. Hartmann (2015), Equivariant motivic integration on formal schemes and the motivic zeta function. arXiv:1511.08656
Zurück zum Zitat F. Heinloth (2007), A note on functional equations for zeta functions with values in Chow motives. Ann. Inst. Fourier (Grenoble) 57(6), 1927–1945MathSciNetMATH F. Heinloth (2007), A note on functional equations for zeta functions with values in Chow motives. Ann. Inst. Fourier (Grenoble) 57(6), 1927–1945MathSciNetMATH
Zurück zum Zitat E. Hrushovski, F. Loeser (2015), Monodromy and the Lefschetz fixed point formula. Ann. Sci. Éc. Norm. Supér. (4) 48(2), 313–349MathSciNetMATH E. Hrushovski, F. Loeser (2015), Monodromy and the Lefschetz fixed point formula. Ann. Sci. Éc. Norm. Supér. (4) 48(2), 313–349MathSciNetMATH
Zurück zum Zitat 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, Paris 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, Paris
Zurück zum Zitat 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. 237–249 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. 237–249
Zurück zum Zitat S. Ishii, A.J. Reguera (2013), Singularities with the highest Mather minimal log discrepancy. Math. Z. 275(3–4), 1255–1274MathSciNetMATHCrossRef S. Ishii, A.J. Reguera (2013), Singularities with the highest Mather minimal log discrepancy. Math. Z. 275(3–4), 1255–1274MathSciNetMATHCrossRef
Zurück zum Zitat T. Ito (2004), Stringy Hodge numbers and p-adic Hodge theory. Compos. Math. 140(6), 1499–1517MathSciNetMATH T. Ito (2004), Stringy Hodge numbers and p-adic Hodge theory. Compos. Math. 140(6), 1499–1517MathSciNetMATH
Zurück zum Zitat 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. France 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. France
Zurück zum Zitat F. Ivorra, J. Sebag (2012), Géométrie algébrique par morceaux, K-équivalence et motifs. Enseign. Math. (2), 58, 375–403MathSciNetMATH F. Ivorra, J. Sebag (2012), Géométrie algébrique par morceaux, K-équivalence et motifs. Enseign. Math. (2), 58, 375–403MathSciNetMATH
Zurück zum Zitat F. Ivorra, J. Sebag (2013), Nearby motives and motivic nearby cycles. Selecta Math. (N.S.) 19(4), 879–902MathSciNetMATH F. Ivorra, J. Sebag (2013), Nearby motives and motivic nearby cycles. Selecta Math. (N.S.) 19(4), 879–902MathSciNetMATH
Zurück zum Zitat J.M. Johnson, J. Kollár (2013), Arc spaces of cA-type singularities. J. Singul. 7, 238–252MathSciNetMATH J.M. Johnson, J. Kollár (2013), Arc spaces of cA-type singularities. J. Singul. 7, 238–252MathSciNetMATH
Zurück zum Zitat M. Kapranov (2000), The elliptic curve in the S-duality theory and Eisenstein series for Kac-Moody groups. arXiv:math/0001005 M. Kapranov (2000), The elliptic curve in the S-duality theory and Eisenstein series for Kac-Moody groups. arXiv:math/0001005
Zurück zum Zitat 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) 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)
Zurück zum Zitat J. Kollár (1996), Rational Curves on Algebraic Varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 32 (Springer, Berlin) J. Kollár (1996), Rational Curves on Algebraic Varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 32 (Springer, Berlin)
Zurück zum Zitat 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 original 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 original
Zurück zum Zitat M. Larsen, V.A. Lunts (2003), Motivic measures and stable birational geometry. Mosc. Math. J. 3(1), 85–95, 259. arXiv:math.AG/0110255MathSciNetMATH M. Larsen, V.A. Lunts (2003), Motivic measures and stable birational geometry. Mosc. Math. J. 3(1), 85–95, 259. arXiv:math.AG/0110255MathSciNetMATH
Zurück zum Zitat M. Larsen, V.A. Lunts (2004), Rationality criteria for motivic zeta functions. Compos. Math. 140(6), 1537–1560MathSciNetMATH M. Larsen, V.A. Lunts (2004), Rationality criteria for motivic zeta functions. Compos. Math. 140(6), 1537–1560MathSciNetMATH
Zurück zum Zitat G. Laumon (1981), Comparaison de caractéristiques d’Euler-Poincaré en cohomologie l-adique. C. R. Acad. Sci. Paris Sér. I Math. 292(3), 209–212MathSciNetMATH G. Laumon (1981), Comparaison de caractéristiques d’Euler-Poincaré en cohomologie l-adique. C. R. Acad. Sci. Paris Sér. I Math. 292(3), 209–212MathSciNetMATH
Zurück zum Zitat 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), 273–287MathSciNetMATH 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), 273–287MathSciNetMATH
Zurück zum Zitat D. Litt (2015), Zeta functions of curves with no rational points. Michigan Math. J. 64(2), 383–395. arXiv:1405.7380MathSciNetMATH D. Litt (2015), Zeta functions of curves with no rational points. Michigan Math. J. 64(2), 383–395. arXiv:1405.7380MathSciNetMATH
Zurück zum Zitat F. Loeser, J. Sebag (2003), Motivic integration on smooth rigid varieties and invariants of degenerations. Duke Math. J. 119(2), 315–344MathSciNetMATH F. Loeser, J. Sebag (2003), Motivic integration on smooth rigid varieties and invariants of degenerations. Duke Math. J. 119(2), 315–344MathSciNetMATH
Zurück zum Zitat I.G. Macdonald (1962), The Poincaré polynomial of a symmetric product. Proc. Camb. Philos. Soc. 58, 563–568MATH I.G. Macdonald (1962), The Poincaré polynomial of a symmetric product. Proc. Camb. Philos. Soc. 58, 563–568MATH
Zurück zum Zitat 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) 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)
Zurück zum Zitat 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 Frenkel 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 Frenkel
Zurück zum Zitat M. Mustaţă (2002), Singularities of pairs via jet schemes. J. Am. Math. Soc. 15(3), 599–615 [electronic] M. Mustaţă (2002), Singularities of pairs via jet schemes. J. Am. Math. Soc. 15(3), 599–615 [electronic]
Zurück zum Zitat J.F. Nash Jr. (1995/1996), Arc structure of singularities. Duke Math. J. 81(1), 31–38. A celebration of John F. Nash, Jr. J.F. Nash Jr. (1995/1996), Arc structure of singularities. Duke Math. J. 81(1), 31–38. A celebration of John F. Nash, Jr.
Zurück zum Zitat J. Nicaise (2009), A trace formula for rigid varieties, and motivic Weil generating series for formal schemes. Math. Ann. 343(2), 285–349MathSciNetMATH J. Nicaise (2009), A trace formula for rigid varieties, and motivic Weil generating series for formal schemes. Math. Ann. 343(2), 285–349MathSciNetMATH
Zurück zum Zitat J. Nicaise (2011b), A trace formula for varieties over a discretely valued field. J. Reine Angew. Math. 650, 193–238MathSciNetMATH J. Nicaise (2011b), A trace formula for varieties over a discretely valued field. J. Reine Angew. Math. 650, 193–238MathSciNetMATH
Zurück zum Zitat J. Nicaise, J. Sebag (2007a), Motivic Serre invariants of curves. Manuscripta Math. 123(2), 105–132MathSciNetMATH J. Nicaise, J. Sebag (2007a), Motivic Serre invariants of curves. Manuscripta Math. 123(2), 105–132MathSciNetMATH
Zurück zum Zitat J. Nicaise, J. Sebag (2007b), Motivic Serre invariants, ramification, and the analytic Milnor fiber. Invent. Math. 168(1), 133–173MathSciNetMATH J. Nicaise, J. Sebag (2007b), Motivic Serre invariants, ramification, and the analytic Milnor fiber. Invent. Math. 168(1), 133–173MathSciNetMATH
Zurück zum Zitat J. Nicaise, C. Xu (2016), Poles of maximal order of motivic zeta functions. Duke Math. J. 165(2), 217–243MathSciNetMATH J. Nicaise, C. Xu (2016), Poles of maximal order of motivic zeta functions. Duke Math. J. 165(2), 217–243MathSciNetMATH
Zurück zum Zitat A.J. Reguera (2006), A curve selection lemma in spaces of arcs and the image of the Nash map. Compos. Math. 142(1), 119–130MathSciNetMATHCrossRef A.J. Reguera (2006), A curve selection lemma in spaces of arcs and the image of the Nash map. Compos. Math. 142(1), 119–130MathSciNetMATHCrossRef
Zurück zum Zitat J. Schepers (2006), Stringy E-functions of varieties with A-D-E singularities. Manuscripta Math. 119(2), 129–157MathSciNetMATH J. Schepers (2006), Stringy E-functions of varieties with A-D-E singularities. Manuscripta Math. 119(2), 129–157MathSciNetMATH
Zurück zum Zitat 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, 14 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, 14
Zurück zum Zitat J. Schepers, W. Veys (2009), Stringy E-functions of hypersurfaces and of Brieskorn singularities. Adv. Geom. 9(2), 199–217MathSciNetMATH J. Schepers, W. Veys (2009), Stringy E-functions of hypersurfaces and of Brieskorn singularities. Adv. Geom. 9(2), 199–217MathSciNetMATH
Zurück zum Zitat A. Smeets (2017), Logarithmic good reduction, monodromy and the rational volume. Algebra & Number Theory 11(1), 213–233MathSciNetMATH A. Smeets (2017), Logarithmic good reduction, monodromy and the rational volume. Algebra & Number Theory 11(1), 213–233MathSciNetMATH
Zurück zum Zitat M. Temkin (2008), Desingularization of quasi-excellent schemes in characteristic zero. Adv. Math. 219(2), 488–522MathSciNetMATH M. Temkin (2008), Desingularization of quasi-excellent schemes in characteristic zero. Adv. Math. 219(2), 488–522MathSciNetMATH
Zurück zum Zitat M. Temkin (2009), Functorial desingularization over \(\mathfrak{q}\): boundaries and the embedded case. Arxiv:0912.2570 M. Temkin (2009), Functorial desingularization over \(\mathfrak{q}\): boundaries and the embedded case. Arxiv:0912.2570
Zurück zum Zitat A.N. Varchenko (1982), The complex singularity index does not change along the stratum μ = const. Funktsional. Anal. i Prilozhen. 16(1), 1–12, 96MathSciNetMATH A.N. Varchenko (1982), The complex singularity index does not change along the stratum μ = const. Funktsional. Anal. i Prilozhen. 16(1), 1–12, 96MathSciNetMATH
Zurück zum Zitat T. Yasuda (2004), Twisted jets, motivic measures and orbifold cohomology. Compos. Math. 140(2), 396–422MathSciNetMATH T. Yasuda (2004), Twisted jets, motivic measures and orbifold cohomology. Compos. Math. 140(2), 396–422MathSciNetMATH
Zurück zum Zitat O. Zariski (1939), The reduction of the singularities of an algebraic surface. Ann. Math. (2) 40, 639–689MathSciNetMATH O. Zariski (1939), The reduction of the singularities of an algebraic surface. Ann. Math. (2) 40, 639–689MathSciNetMATH
Zurück zum Zitat Z. Zhu (2013), Log canonical thresholds in positive characteristic. arXiv:1308.5445 Z. Zhu (2013), Log canonical thresholds in positive characteristic. arXiv:1308.5445
Metadaten
Titel
Applications
verfasst von
Antoine Chambert-Loir
Johannes Nicaise
Julien Sebag
Copyright-Jahr
2018
Verlag
Springer New York
DOI
https://doi.org/10.1007/978-1-4939-7887-8_7