# Virtual rigid motives of semi-algebraic sets

## TL;DR

This paper constructs a new ring morphism linking semi-algebraic sets over a field to motives of rigid analytic varieties, extending classical motives and connecting with motivic integration and duality, thereby answering a key question about the analytic Milnor fiber.

## Contribution

It introduces a novel ring morphism from semi-algebraic sets to motives of rigid analytic varieties, extending classical motives and integrating with existing motivic frameworks.

## Key findings

- Established a ring morphism extending algebraic motives to semi-algebraic sets.
- Connected the construction with motivic integration and duality principles.
- Provided an answer to a question about the analytic Milnor fiber.

## Abstract

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 group of semi-algebraic sets over $K$ to the Grothendieck group of motives of rigid analytic varieties over $K$. It extend 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.