Motivic integration and Milnor fiber. (English) Zbl 1498.14032

Fichou, Goulwen; Yin, Yimu

J. Eur. Math. Soc. (JEMS) 24, No. 5, 1617-1678 (2022).

Summary: We put forward a uniform narrative that weaves together several variants of Hrushovski-Kazhdan style integral, and describe how it can facilitate the understanding of the Denef-Loeser motivic Milnor fiber and closely related objects. Our study focuses on the so-called “nonarchimedean Milnor fiber” that was introduced by Hrushovski and Loeser, and our thesis is that it is a richer embodiment of the underlying philosophy of the Milnor construction. The said narrative is first developed in the more natural complex environment, and is then extended to the real one via descent. In the process of doing so, we are able to provide more illuminating new proofs, free of resolution of singularities, of a few pivotal results in the literature, both complex and real. To begin with, the real motivic zeta function is shown to be rational, which yields the real motivic Milnor fiber; this is an analogue of the Hrushovski-Loeser construction. Then, applying \(T\)-convex integration after descent, matching the Euler characteristics of the topological Milnor fiber and the motivic Milnor fiber becomes a matter of simple computation, which is not only free of resolution of singularities as in the Hrushovski-Loeser proof, but is also free of other sophisticated algebro-geometric machineries. Finally, we also establish, in a much more intuitive manner, a new Thom-Sebastiani formula, which can be specialized to the one given by Guibert-Loeser-Merle.

MSC:

14E18	Arcs and motivic integration
12J25	Non-Archimedean valued fields
14B05	Singularities in algebraic geometry

Keywords:

Hrushovski-Kazhdan style motivic integration; equivariant Grothendieck ring; motivic zeta function; Denef-Loeser motivic Milnor fiber; Thom-Sebastiani formula; \(T\)-convex valued field

References:

[1]	References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1677 References [1] Abramovich, D., Karu, K., Matsuki, K., Włodarczyk, J.: Torification and factorization of bira-tional maps. J. Amer. Math. Soc. 15, 531-572 (2002) Zbl 1032.14003 MR 1896232 · Zbl 1032.14003
[2]	Bittner, F.: The universal Euler characteristic for varieties of characteristic zero. Compos. Math. 140, 1011-1032 (2004) Zbl 1086.14016 MR 2059227 · Zbl 1086.14016
[3]	Bittner, F.: On motivic zeta functions and the motivic nearby fiber. Math. Z. 249, 63-83 (2005) Zbl 1085.14020 MR 2106970 · Zbl 1085.14020
[4]	Bochnak, J., Coste, M., Roy, M.-F.: Real Algebraic Geometry. Ergeb. Math. Grenzgeb. (3) 36, Springer, Berlin (1998) Zbl 0912.14023 MR 1659509 · Zbl 0912.14023
[5]	Campesato, J.-B.: On a motivic invariant of the arc-analytic equivalence. Ann. Inst. Fourier (Grenoble) 67, 143-196 (2017) Zbl 1408.14186 MR 3612328 · Zbl 1408.14186
[6]	Comte, G., Fichou, G.: Grothedieck ring of semialgebraic formulas and motivic real Milnor fibers. Geom. Topol. 18, 963-996 (2014) Zbl 1388.14159 MR 3190606 · Zbl 1388.14159
[7]	Coste, M.: Real algebraic sets. In: Arc Spaces and Additive Invariants in Real Algebraic and Analytic Geometry, Panor. Synthèses 24, Soc. Math. France, Paris, 1-32 (2007) Zbl 1144.14305 MR 2409684 · Zbl 1144.14305
[8]	Delfs, H., Knebusch, M.: On the homology of algebraic varieties over real closed fields. J. Reine Angew. Math. 335, 122-163 (1982) Zbl 0484.14006 MR 667464 · Zbl 0484.14006
[9]	Denef, J., Loeser, F.: Motivic exponential integrals and a motivic Thom-Sebastiani theorem. Duke Math. J. 99, 285-309 (1999) Zbl 0966.14015 MR 1708026 · Zbl 0966.14015
[10]	Denef, J., Loeser, F.: Geometry on arc spaces of algebraic varieties. In: European Congress of Mathematics, Vol. I (Barcelona, 2000), Progr. Math. 201, Birkhäuser, Basel, 327-348 (2001) Zbl 1079.14003 MR 1905328 · Zbl 1079.14003
[11]	Denef, J., Loeser, F.: Lefschetz numbers of iterates of the monodromy and truncated arcs. Topology 41, 1031-1040 (2002) Zbl 1054.14003 MR 1923998 · Zbl 1054.14003
[12]	van den Dries, L.: Tame Topology and o-Minimal Structures. London Math. Soc. Lecture Note Ser. 248, Cambridge Univ. Press, Cambridge (1998) Zbl 0953.03045 MR 1633348 · Zbl 0953.03045
[13]	van den Dries, L., Lewenberg, A. H.: T -convexity and tame extensions. J. Symbolic Logic 60, 74-102 (1995) Zbl 0856.03028 MR1324502 · Zbl 0856.03028
[14]	Engler, A. J., Prestel, A.: Valued Fields. Springer Monogr. Math., Springer, Berlin (2005) Zbl 1128.12009 MR 2183496 · Zbl 1128.12009
[15]	Fichou, G.: Motivic invariants of arc-symmetric sets and blow-Nash equivalence. Compos. Math. 141, 655-688 (2005) Zbl 1080.14070 MR 2135282 · Zbl 1080.14070
[16]	Fichou, G.: Equivariant virtual Betti numbers. Ann. Inst. Fourier (Grenoble) 58, 1-27 (2008) Zbl 1142.14003 MR 2401214 · Zbl 1142.14003
[17]	Fichou, G.: The motivic real Milnor fibres. Manuscripta Math. 139, 167-178 (2012) Zbl 1258.14002 MR 2959676 · Zbl 1258.14002
[18]	Fichou, G.: On Grothendieck rings and algebraically constructible functions. Math. Ann. 369, 761-795 (2017) Zbl 1423.14060 MR 3694660 · Zbl 1423.14060
[19]	Fichou, G., Shiota, M.: Real Milnor fibres and Puiseux series. Ann. Sci. École Norm. Sup. (4) 50, 1205-1240 (2017) Zbl 1386.14199 MR 3720028 · Zbl 1386.14199
[20]	Forey, A., Yin, Y.: Bounded integral and motivic Milnor fiber. arXiv:1910.12764v2 (2019)
[21]	Guibert, G., Loeser, F., Merle, M.: Nearby cycles and composition with a nondegenerate poly-nomial. Int. Math. Res. Notices 2005, 1873-1888 Zbl 1093.14032 MR 2171196 · Zbl 1093.14032
[22]	Guibert, G., Loeser, F., Merle, M.: Iterated vanishing cycles, convolution, and a motivic ana-logue of a conjecture of Steenbrink. Duke Math. J. 132, 409-457 (2006) Zbl 1173.14301 MR 2219263 · Zbl 1173.14301
[23]	Guibert, G., Loeser, F., Merle, M.: Composition with a two variable function. Math. Res. Lett. 16, 439-448 (2009) Zbl 1187.14046 MR 2511624 · Zbl 1187.14046
[24]	Hrushovski, E., Kazhdan, D.: Integration in valued fields. In: Algebraic Geometry and Number Theory, Progr. Math. 253, Birkhäuser Boston, Boston, MA, 261-405 (2006) Zbl 1136.03025 MR 2263194 · Zbl 1136.03025
[25]	Hrushovski, E., Loeser, F.: Monodromy and the Lefschetz fixed point formula. Ann. Sci. École Norm. Sup. (4) 48, 313-349 (2015) Zbl 1400.14015 MR 3346173 · Zbl 1400.14015
[26]	Koike, S., Parusiński, A.: Motivic-type invariants of blow-analytic equivalence. Ann. Inst. Fourier (Grenoble) 53, 2061-2104 (2003) Zbl 1062.14006 MR 2044168 · Zbl 1062.14006
[27]	Lê, Q. T.: The motivic Thom-Sebastiani theorem for regular and formal functions. J. Reine Angew. Math. 735, 175-198 (2018) Zbl 06836106 MR 3757474 · Zbl 1460.14037
[28]	Looijenga, E.: Motivic measures. Astérisque 276, 267-297 (2002) Zbl 0996.14011 MR 1886763 · Zbl 0996.14011
[29]	McCrory, C., Parusiński, A.: Complex monodromy and the topology of real algebraic sets. Compos. Math. 106, 211-233 (1997) Zbl 0949.14037 MR 1457340 · Zbl 0949.14037
[30]	McCrory, C., Parusiński, A.: Virtual Betti numbers of real algebraic varieties. C. R. Math. Acad. Sci. Paris 336, 763-768 (2003) Zbl 1073.14071 MR 1989277 · Zbl 1073.14071
[31]	Nicaise, J., Payne, S.: A tropical motivic Fubini theorem with applications to Donaldson-Thomas theory. Duke Math. J. 168, 1843-1886 (2019) Zbl 1429.14033 MR 3983293 · Zbl 1429.14033
[32]	Nicaise, J., Payne, S., Schroeter, F.: Tropical refined curve counting via motivic integration. Geom. Topol. 22, 3175-3234 (2018) Zbl 1430.14037 MR 3858763 · Zbl 1430.14037
[33]	Yin, Y.: Special transformations in algebraically closed valued fields. Ann. Pure Appl. Logic 161, 1541-1564 (2010) Zbl 1226.03044 MR 2674050 · Zbl 1226.03044
[34]	Yin, Y.: Generalized Euler characteristic in power-bounded T -convex valued fields. Compos. Math. 153, 2591-2642 (2017) Zbl 06810461 MR 3705299 · Zbl 1506.03093