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The motivic Thom–Sebastiani theorem for regular and formal functions

  • Quy Thuong Lê EMAIL logo

Abstract

Thanks to the work of Hrushovski and Loeser on motivic Milnor fibers, we give a model-theoretic proof for the motivic Thom–Sebastiani theorem in the case of regular functions. Moreover, slightly extending Hrushovski–Loeser’s construction adjusted to Sebag, Loeser and Nicaise’s motivic integration for formal schemes and rigid varieties, we formulate and prove an analogous result for formal functions. The latter is meaningful as it has been a crucial element of constructing Kontsevich–Soibelman’s theory of motivic Donaldson–Thomas invariants.

Award Identifier / Grant number: 246903/NMNAG

Funding statement: The author is partially supported by the Centre Henri Lebesgue (program “Investissements d’avenir”, ANR-11-LABX-0020-01) and by ERC under the European Community’s Seventh Framework Programme (FP7/2007-2013), ERC Grant Agreement no. 246903/NMNAG.

Acknowledgements

The author is grateful to François Loeser, Julien Sebag and Michel Raibaut for useful discussions. He would like to thank the Centre Henri Lebesgue and the Université de Rennes 1 for awarding him a postdoctoral fellowship and an excellent atmosphere during his stay there.

References

[1] A. Abbes, Éléments de géométrie rigide. Volume I: Construction et étude géométrique des espaces rigides, Progr. Math. 286, Birkhäuser, Basel 2010. 10.1007/978-3-0348-0012-9Search in Google Scholar

[2] S. Bosch, W. Lütkebohmert and M. Raynaud, Formal and rigid geometry, III: The relative maximum principle, Math. Ann. 302 (1995), 1–29. 10.1007/BF01444485Search in Google Scholar

[3] J. Denef and F. Loeser, Motivic Igusa zeta functions, J. Algebraic Geom. 7 (1998), 505–537. Search in Google Scholar

[4] J. Denef and F. Loeser, Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math. 135 (1999), 201–232. 10.1007/s002220050284Search in Google Scholar

[5] J. Denef and F. Loeser, Motivic exponential integrals and a motivic Thom–Sebastiani theorem, Duke Math. J. 99 (1999), 285–309. 10.1215/S0012-7094-99-09910-6Search in Google Scholar

[6] J. Denef and F. Loeser, Geometry on arc spaces of algebraic varieties, European congress of mathematics. Vol. 1 (Barcelona 2000), Progr. Math. 201, Birkhäuser, Basel (2001), 327–348. 10.1007/978-3-0348-8268-2_19Search in Google Scholar

[7] J. Denef and F. Loeser, Lefschetz numbers of iterates of the monodromy and truncated arcs, Topology 41 (2002), 1031–1040. 10.1016/S0040-9383(01)00016-7Search in Google Scholar

[8] L. Fu, A Thom–Sebastiani theorem in characteristic p, Math. Res. Lett. 21 (2014), 101–119. 10.4310/MRL.2014.v21.n1.a8Search in Google Scholar

[9] M. J. Greenberg, Schemata over local rings, Ann. of Math. 73 (1961), 624–648. 10.2307/1970321Search in Google Scholar

[10] G. Guibert, F. Loeser and M. Merle, Nearby cycles and composition with a non-degenerate polynomial, Int. Math. Res. Not. IMRN 31 (2005), 1873–1888. 10.1155/IMRN.2005.1873Search in Google Scholar

[11] G. Guibert, F. Loeser and M. Merle, Iterated vanishing cycles, convolution, and a motivic analogue of the conjecture of Steenbrink, Duke Math. J. 132 (2006), 409–457. 10.1215/S0012-7094-06-13232-5Search in Google Scholar

[12] G. Guibert, F. Loeser and M. Merle, Composition with a two variable function, Math. Res. Lett. 16 (2009), 439–448. 10.4310/MRL.2009.v16.n3.a5Search in Google Scholar

[13] E. Hrushovski and D. Kazhdan, Integration in valued fields, Algebraic geometry and number theory, Progr. Math. 253, Birkhäuser, Basel (2006), 261–405. 10.1007/978-0-8176-4532-8_4Search in Google Scholar

[14] E. Hrushovski and F. Loeser, Monodromy and the Lefschetz fixed point formula, Ann. Sci. École Norm. Sup. 48 (2015), 313–349. 10.24033/asens.2246Search in Google Scholar

[15] M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson–Thomas invariants and cluster transformations, preprint (2008), http://arxiv.org/abs/0811.2435. Search in Google Scholar

[16] G. Laumon, Transformation de Fourier, constantes d’équations fontionnelles, et conjecture de Weil, Publ. Math. IHES 65 (1987), 131–210. 10.1007/BF02698937Search in Google Scholar

[17] Q. T. Lê, Proofs of the integral identity conjecture over algebraically closed fields, Duke Math. J. 164 (2015), 157–194. 10.1215/00127094-2869138Search in Google Scholar

[18] F. Loeser and J. Sebag, Motivic integration on smooth rigid varieties and invariants of degenerations, Duke Math. J. 119 (2003), 315–344. 10.1215/S0012-7094-03-11924-9Search in Google Scholar

[19] E. Looijenga, Motivic measures, Astérisque 276 (2002), 267–297. Search in Google Scholar

[20] J. Nicaise, A trace formula for rigid varieties, and motivic Weil generating series for formal schemes, Math. Ann. 343 (2009), 285–349. 10.1007/s00208-008-0273-9Search in Google Scholar

[21] J. Nicaise and J. Sebag, Motivic Serre invariants, ramification, and the analytic Milnor fiber, Invent. Math. 168 (2007), 133–173. 10.1007/s00222-006-0029-7Search in Google Scholar

[22] M. Saito, Mixed Hodge modules and applications, Proceedings of the ICM (Kyoto 1990), Springer, Tokyo (1991), 725–734. Search in Google Scholar

[23] J. Sebag, Intégration motivique sur les schémas formels, Bull. Soc. Math. France 132 (2004), 1–54. 10.24033/bsmf.2458Search in Google Scholar

[24] M. Sebastiani and R. Thom, Un résultat sur la monodromie, Invent. Math. 13 (1971), 90–96. 10.1007/BF01390095Search in Google Scholar

[25] J. H. M. Steenbrink, Mixed Hodge structure on the vanishing cohomology, Real and complex singularities (Oslo 1976), Sijthoff and Noordhoff, Alphen aan den Rijn (1977), 525–563. 10.1007/978-94-010-1289-8_15Search in Google Scholar

[26] M. Temkin, Desingularization of quasi-excellent schemes in characteristic zero, Adv. Math. 219 (2008), 488–522. 10.1016/j.aim.2008.05.006Search in Google Scholar

[27] A. N. Varchenko, Asymptotic Hodge structure in the vanishing cohomology (Russian), Izv. Akad. Nauk SSSR Ser. Math. 45 (1981), 540–591; translation in Math. USSR Izv. 18 (1982), 469–512. Search in Google Scholar

Received: 2014-5-28
Published Online: 2015-6-16
Published in Print: 2018-2-1

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