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SUMMARY:The motivic McKay correspondence in positive and mixed characteris
tics
DTSTART;VALUE=DATE-TIME:20151118T133000Z
DTEND;VALUE=DATE-TIME:20151118T143000Z
DTSTAMP;VALUE=DATE-TIME:20210622T070416Z
UID:indico-event-901@indico.math.cnrs.fr
DESCRIPTION:« Return of the IHÉS Postdoc Seminar »\n\n \n\nAbstract: I
n this talk\, I will talk about the motivic McKay correspondence studied b
y Batyrev and Denef-Loeser and its generalization to positive and mixed ch
aracteristics by myself partly in collaboration with M.M. Wood. The latter
relates stringy invariants of singularities to weighted counts of Galois
extensions of a local field.\n\nFor a finite dimensional linear representa
tion of a finite group\, the motivic McKay correspondence says that the mo
tivic stringy invariant of the associated quotient variety is equal to a f
inite sum of classes of affine spaces in a certain modification of the Gro
thendieck ring of varieties.\n\nIn order to understand wild quotient singu
larities\, which are known to be typical “bad” singularities in positi
ve/mixed characteristics\, I started an attempt to generalize the motivic
McKay correspondence to positive and mixed characteristics and formulated
a conjectural generalization. Here\, the finite sum of classes of affine v
arieties is replaced with a motivic integral over the (conjectural) moduli
space of G-covers of formal discs. When the base field (or the residue fi
eld if working over a complete DVR) is finite\, then the point counting re
alization of this motivic integral is a weighted count of Galois extension
s of the power series field (or the fraction field of the DVR). Such count
s were previously studied by number-theorists including Krasner\, Serre\,
Bhargava\, Kedlaya and Wood. The point counting version of the McKay corre
spondence was recently proved by myself. A part of my motivation of this w
ork is a search for a counterexample of resolution of singularities\, whic
h has not been successful till now.\n\nIf time allows\, I will also explai
n that from this result and a heuristic argument\, one can relate Malle’
s conjecture on distribution of Galois extensions of number fields and Man
in’s conjecture on distribution of rational points on Fano varieties.\n\
nhttps://indico.math.cnrs.fr/event/901/
LOCATION:IHES Amphithéâtre Léon Motchane
URL:https://indico.math.cnrs.fr/event/901/
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