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.


Advisor(s):
Hausel, Tamás
Year:
2017
Publisher:
Lausanne, EPFL
Keywords:
Other identifiers:
urn: urn:nbn:ch:bel-epfl-thesis8025-1
Laboratories:




 Record created 2017-10-27, last modified 2019-05-09

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