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On a logarithmic version of the derived McKay correspondence

Part of: Families, fibrations (Co)homology theory Birational geometry

Published online by Cambridge University Press:  08 November 2018

Sarah Scherotzke ,
Nicolò Sibilla  and
Mattia Talpo
Sarah Scherotzke
Affiliation:
Mathematical Institute of the University of Münster, Einsteinstrasse 62, 48149 Münster, Germany email scherotz@math.uni-muenster.de
Nicolò Sibilla
Affiliation:
Max Planck Institute for Mathematics, Bonn, Germany School of Mathematics, Statistics and Actuarial Sciences, University of Kent, Canterbury, Kent CT2 7NF, UK email N.Sibilla@kent.ac.uk
Mattia Talpo
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK email m.talpo@imperial.ac.uk
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Abstract

We globalize the derived version of the McKay correspondence of Bridgeland, King and Reid, proven by Kawamata in the case of abelian quotient singularities, to certain logarithmic algebraic stacks with locally free log structure. The two sides of the correspondence are given respectively by the infinite root stack and by a certain version of the valuativization (the projective limit of every possible logarithmic blow-up). Our results imply, in particular, that in good cases the category of coherent parabolic sheaves with rational weights is invariant under logarithmic blow-up, up to Morita equivalence.

Keywords

McKay correspondencesemistable degenerationsparabolic sheaves

MSC classification

Primary: 14F05: Sheaves, derived categories of sheaves and related constructions 14D06: Fibrations, degenerations 14E16: McKay correspondence

Information

Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 12 , December 2018 , pp. 2534 - 2585
Copyright
© The Authors 2018 

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