# Virtual rigid motives of semi-algebraic sets

Authors
• 1 ETH Zürich, D-Math, Rämistrasse 101, Zurich, 8092, Switzerland , Zurich (Switzerland)
Type
Published Article
Journal
Selecta Mathematica
Publisher
Springer International Publishing
Publication Date
Feb 06, 2019
Volume
25
Issue
1
Identifiers
DOI: 10.1007/s00029-019-0453-3
Source
Springer Nature
Keywords
Let k be a field of characteristic zero containing all roots of unity and K=k((t))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K=k(( t))$$\end{document}. 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.