Virtual rigid motives of semi-algebraic sets
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Let k be a field of characteristic zero containing all roots of unity and \(K=k(( t))\). We build a ring morphism from the Grothendieck ring of semi-algebraic sets over K to the Grothendieck ring of motives of rigid analytic varieties over K. It extends the morphism sending the class of an algebraic variety over K to its cohomological motive with compact support. We show that it fits inside a commutative diagram involving Hrushovski and Kazhdan’s motivic integration and Ayoub’s equivalence between motives of rigid analytic varieties over K and quasi-unipotent motives over k; we also show that it satisfies a form of duality. This allows us to answer a question by Ayoub, Ivorra and Sebag about the analytic Milnor fiber.
KeywordsMotivic integration Rigid motives Rigid analytic geometry Motivic Milnor fiber Analytic Milnor fiber
Mathematics Subject Classification14C15 14F42 03C60 14G22 32S30
I would like to thank François Loeser for his constant support during the preparation of this project. I also thank Joseph Ayoub for a very careful reading that greatly helped to improve this paper, and Marco Robalo, Florian Ivorra and Julien Sebag for discussions and remarks. Thanks also to the referee for valuable remarks that helped to improve this paper. This research was partially supported by ANR-15-CE40-0008 (Défigéo).
- 5.Berthelot, P.: Cohomologie Rigide et Cohomologie Rigide à Supports Propres, Première Partie. Prépublication IRMAR 96-03, Université de Rennes, (1996)Google Scholar
- 11.Cisinski, D.-C., Déglise, F.: Triangulated categories of mixed motives. arXiv:0912.2110 [math] (2009)
- 13.Deligne, P.: Voevodsky’s Lectures on Cross Functors (2001)Google Scholar
- 16.Denef, J., Loeser, F.: Geometry on arc spaces of algebraic varieties. In: European Congress of Mathematics, Vol. I (Barcelona, 2000), volume 201 of Progr. Math., pp. 327–348. Birkhäuser, Basel (2001)Google Scholar
- 20.Hrushovski, E., Kazhdan, D.: Integration in valued fields. In: Algebraic Geometry and Number Theory. In Honor of Vladimir Drinfeld’s 50th Birthday, pp. 261–405. Birkhäuser, Basel (2006)Google Scholar
- 25.Loeser, F.: Seattle lectures on motivic integration. In: Algebraic Geometry, Seattle 2005. Proceedings of the 2005 Summer Research Institute, Seattle, WA, USA, July 25–August 12, 2005, pp. 745–784. American Mathematical Society (AMS), Providence, RI (2009)Google Scholar
- 28.Nicaise, J., Payne, S.: A tropical motivic Fubini theorem with applications to Donaldson–Thomas theory. Duke. Math. J. arXiv:1703.10228 (2017) (to appear)
- 31.Stacks project authors. The Stacks project (2018)Google Scholar