A Proof of the Integral Identity via Braden's Theorem

TL;DR

This paper presents a concise proof of a generalized integral identity for virtual motives within the motivic stable homotopy category, utilizing Braden's hyperbolic localization theorem in a novel context.

Contribution

It extends Braden's theorem to motivic sheaves and provides a simplified proof of a conjectured integral identity by Kontsevich and Soibelman.

Findings

Proof of the integral identity using motivic homotopy theory

Extension of Braden's theorem to motivic sheaves

Simplification of the proof of a conjecture in virtual motives

Abstract

The purpose of this paper is to provide a very short proof of a generalized categorified version, within the motivic stable homotopy category of Morel and Voevodsky, of the integral identity for virtual motives conjectured by Kontsevich and Soibelman. Our proof is an application of an important result in geometric representation theory due to Braden and known as the hyperbolic localization/restriction theorem. Though originally proved in the context of etale sheaves (or sheaves on the associated complex analytic space in the case of complex algebraic varieties) Braden's theorem turns out to hold also in the context of motivic sheaves, at least in the special case of vector bundles with a linear G_m-action.