## Motivic and p-adic Localization Phenomena

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 $\FI_\mu(s)$ of $\mu$, and in particular to a small set of candidate poles for $\FI_\mu(s)$. We also study various properties of the residue at the largest pole of $\FI_\mu(s)$. 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.

Directeur(s):
Hausel, Tamás
Année
2017
Publisher:
Lausanne, EPFL
Mots-clefs:
Autres identifiants:
urn: urn:nbn:ch:bel-epfl-thesis8025-1
Laboratoires:

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Notice créée le 2017-10-27, modifiée le 2019-12-05

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