Skip to main content
SpringerLink
Log in

On Grothendieck rings and algebraically constructible functions

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

We investigate Grothendieck rings appearing in real geometry, notably for arc-symmetric sets, and focus on the relative case in analogy with the properties of the ring of algebraically constructible functions defined by McCrory and Parusiński. We study in particular the duality and link operators, including their behaviour with respect to motivic Milnor fibres with signs.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abramovich, D., Karu, K., Matsuki, K., Wlodarczyk, J.: Torification and factorization of birational maps. J. Am. Math. Soc. 15(3), 531–572 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bittner, F.: The universal Euler characteristic for varieties of characteristic zero. Compos. Math. 140, 1011–1032 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bittner, F.: On motivic zeta functions and the motivic nearby fiber. Math. Z. 249(1), 63–83 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bochnak, J., Coste, M., Roy, M.-F.: Géométrie algébrique réelle, Ergeb. Math. Grenzgeb. 3 Folge, 12. Springer, Berlin (1987)

    Google Scholar 

  5. Bories, B., Veys, W.: Igusa’s \(p\)-adic local zeta function and the monodromy conjecture for non-degenerated surface singularities. Mem. Am. Math. Soc 242, 1145 (2016)

    Google Scholar 

  6. Borisov, L.: The class of the affine line is a zero divisor in the Grothendieck ring (2014). arXiv:1412.6194

  7. Campesato, J.-B.: On a motivic invariant of the arc-analytic equivalence. Annales de l’Institut Fourier 7, 143–196 (2017)

    Article  MathSciNet  Google Scholar 

  8. Campesato, J.-B.: From the blow-analytic equivalence to the arc-analytic equivalence: a survey. In: Proceedings of the Sixth Japanese-Australian Workshop on Real & Complex Singularities, Saitama Math. J. 31, 35–78 (2017)

  9. Campesato, J.-B.: On the arc-analytic type of some weighted homogeneous polynomials (2016). arXiv:1612.08269

  10. Comte, G., Fichou, G.: Grothendieck ring of semialgebraic formulas and motivic real Milnor fibres. Geom. Topol. 18(2), 963–996 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Coste, M.: Real Algebraic Sets, Arc Spaces and Additive Invariants in Real Algebraic and Analytic Geometry, 1–32, Panor. Synthèses, vol. 24. Soc. Math. France, Paris (2007)

    Google Scholar 

  12. Cluckers, R., Loeser, F.: Constructible motivic functions and motivic integration. Invent. Math. 173(1), 23–121 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Denef, J., Loeser, F.: Germs of arcs on singular algebraic varieties and motivic integration. Invent. Math. 135, 201–232 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  14. Deveney, J.K.: Ruled function fields. Proc. Am. Math. Soc. 86(2), 213–215 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fichou, G.: Motivic invariants of arc-symmetric sets and blow-Nash equivalence. Compos. Math. 141(3), 655–688 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fichou, G., Fukui, T.: Motivic invariant of real polynomial functions and Newton polyhedron. Math. Proc. Camb. Philos. Soc. 160(1), 141–166 (2016)

    Article  MathSciNet  Google Scholar 

  17. Fichou, G., Huisman, J., Mangolte, F., Monnier, J.-P.: Fonctions régulues. J. Reine Angew. Math. 718, 103–151 (2016)

    MathSciNet  MATH  Google Scholar 

  18. Fichou, G., Shiota, M.: Virtual Poincaré polynomial of the link of a real algebraic variety. Math. Z. 273(3–4), 1053–1061 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gromov, M.: Endomorphisms of symbolic algebraic varieties. J. Eur. Math. Soc. (JEMS) 1(2), 109–197 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  20. Koike, S., Parusiński, A.: Motivic-type invariants of blow-analytic equivalence. Ann. Inst. Fourier (Grenoble) 53(7), 2061–2104 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kollár, J., Nowak, K.: Continuous rational functions on real and \(p\)-adic varieties. Math. Z. 279(1–2), 85–97 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kucharz, W.: Rational maps in real algebraic geometry. Adv. Geom. 9(4), 517–539 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kurdyka, K.: Ensemble semi-algébriques symétriques par arcs. Math. Ann. 282, 445–462 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kurdyka, K., Parusiński, A.: Arc-symmetric Sets and Arc-analytic Mappings, Arc Spaces and Additive Invariants in Real Algebraic and Analytic Geometry, 33-67, Panor. Synthèses, vol. 24. Soc. Math. France, Paris (2007)

    MATH  Google Scholar 

  25. Larsen, M., Lunts, V.A.: Motivic measures and stable birational geometry. Mosc. Math. J. 3(1), 85–95 (2003)

    MathSciNet  MATH  Google Scholar 

  26. Looijenga, E.: Motivic measures, Séminaire Bourbaki, vol. 1999/2000. Astérisque 276, 267–297 (2002)

    Google Scholar 

  27. McCrory, C., Parusiński, A.: Algebraically constructible functions. Ann. Sci. École Norm. Sup. (4) 30(4), 527–552 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  28. McCrory, C., Parusiński, A.: Virtual Betti numbers of real algebraic varieties. C. R. Math. Acad. Sci. Paris 336(9), 763–768 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  29. McCrory, C., Parusiński, A.: The Weight Filtration for Real Algebraic Varieties, Topology of Stratified Spaces. 121–160, Mathematical Sciences Research Institute Publications, 58, Cambridge University Press, Cambridge (2011)

    MATH  Google Scholar 

  30. Nicaise, J., Sebag, J.: Motivic Serre invariants, ramification, and the analytic Milnor fiber. Invent. Math 168(1), 133–173 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  31. Parusiński, A.: Topology of injective endomorphisms of real algebraic sets. Math. Ann. 328(1–2), 353–372 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  32. Parusiński, A., Paunescu, L.: Arcwise analytic stratification, whitney fibering conjecture and Zariski equisingularity. Adv. Math. 309, 254–305 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  33. Poonen, B.: The Grothendieck ring of varieties is not a domain. Math. Res. Lett. 9(4), 493–497 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  34. Priziac, F.: Equivariant zeta functions for invariant Nash germs. Nagoya M. J. 222, 100–136 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  35. Priziac, F.: On the equivariant blow-Nash classification of simple invariant Nash germs (2016). arXiv:1606.00303

  36. Quarez, R.: Espace des germes d’arcs réels et série de Poincaré d’un ensemble semi-algébrique. Ann. Inst. Fourier (Grenoble) 51(1), 43–68 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  37. Schapira, P.: Operations on constructible functions. J. Pure Appl. Algebra 72(1), 83–93 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  38. Shiota, M.: Nash Manifolds, Lecture Notes in Mathematics. Springer, Berlin (1987)

    Book  Google Scholar 

  39. van den Dries, L.: Tame Topology and O-minimal Structures, London Mathematical Society Lecture Note Series, vol. 248. Cambridge University Press, Cambridge (1998)

    Book  MATH  Google Scholar 

  40. Viro, O.: Some Integral Calculus Based on Euler Characteristic, Topology and Geometry-Rohlin Seminar, 127–138, Lecture Notes in Mathematics, vol. 1346. Springer, Berlin (1988)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. IRMAR (UMR 6625), Université de Rennes 1, Campus de Beaulieu, 35042, Rennes Cedex, France

    Goulwen Fichou

Authors
  1. Goulwen Fichou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Goulwen Fichou.

Additional information

The author wish to thank Jean-Baptiste Campesato, Michel Coste, Toshizumi Fukui, Julia Gordon and Adam Parusiński for useful discussions, and is deeply grateful to the UMI PIMS of the CNRS where this project has been carried out. He has also received support from ANR-15-CE40-0008 (Défigéo).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fichou, G. On Grothendieck rings and algebraically constructible functions. Math. Ann. 369, 761–795 (2017). https://doi.org/10.1007/s00208-017-1564-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-017-1564-9

Search

Navigation