Motivic and $p$-adic Localization Phenomena

TL;DR

This thesis explores motivic classes of certain algebraic varieties and analyzes $p$-adic measures, leading to explicit formulas for Igusa zeta functions and conjectures relating to quiver representations, revealing deep localization phenomena.

Contribution

It provides explicit motivic class computations and $p$-adic formulas for hypertoric and Nakajima varieties, introducing new conjectures on residues linked to quiver representations.

Findings

Explicit motivic classes for hypertoric and Nakajima varieties

Derived formulas for Igusa zeta functions of hypertoric moment maps

Conjectural links between residues and indecomposable quiver representations

Abstract

In this thesis we compute motivic classes of hypertoric varieties, Nakajima quiver varieties and open de Rham spaces in a certain localization of the Grothendieck ring of varieties. Furthermore we study the $p$-adic pushforward of the Haar measure under a hypertoric moment map $\mu$. This leads to an explicit formula for the Igusa zeta function $I_\mu(s)$ of $\mu$, and in particular to a small set of candidate poles for $I_\mu(s)$. We also study various properties of the residue at the largest pole of $I_\mu(s)$. Finally, if $\mu$ is constructed out of a quiver $\Gamma$ we give a conjectural description of this residue in terms of indecomposable representations of $\Gamma$ over finite depth rings. The connections between these different results is the method of proof. At the heart of each theorem lies a motivic or $p$-adic volume computation, which is only possible due to some surprising cancellations. These cancellations are reminiscent of a result in classical symplectic geometry by Duistermaat and Heckman on the localization of the Liouville measure, hence the title of the thesis.