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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Parity sheaves
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by Daniel Juteau, Carl Mautner and Geordie Williamson PDF
J. Amer. Math. Soc. 27 (2014), 1169-1212 Request permission

Abstract:

Given a stratified variety $X$ with strata satisfying a cohomological parity-vanishing condition, we define and show the uniqueness of “parity sheaves,” which are objects in the constructible derived category of sheaves with coefficients in an arbitrary field or complete discrete valuation ring. This construction depends on the choice of a parity function on the strata.

If $X$ admits a resolution also satisfying a parity condition, then the direct image of the constant sheaf decomposes as a direct sum of parity sheaves, and the multiplicities of the indecomposable summands are encoded in certain refined intersection forms appearing in the work of de Cataldo and Migliorini. We give a criterion for the Decomposition Theorem to hold in the semi-small case.

Our framework applies to many stratified varieties arising in representation theory such as generalised flag varieties, toric varieties, and nilpotent cones. Moreover, parity sheaves often correspond to interesting objects in representation theory. For example, on flag varieties we recover in a unified way several well-known complexes of sheaves. For one choice of parity function we obtain the indecomposable tilting perverse sheaves. For another, when using coefficients of characteristic zero, we recover the intersection cohomology sheaves and in arbitrary characteristic the special sheaves of Soergel, which are used by Fiebig in his proof of Lusztig’s conjecture.

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Additional Information
  • Daniel Juteau
  • Affiliation: LMNO, Université de Caen Basse-Normandie, CNRS, BP 5186, 14032 Caen, France
  • Email: daniel.juteau@unicaen.fr
  • Carl Mautner
  • Affiliation: Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany
  • MR Author ID: 1062385
  • Email: cmautner@mpim-bonn.mpg.de
  • Geordie Williamson
  • Affiliation: Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany
  • MR Author ID: 845262
  • Email: geordie@mpim-bonn.mpg.de
  • Received by editor(s): November 5, 2012
  • Received by editor(s) in revised form: October 21, 2013
  • Published electronically: May 21, 2014
  • Additional Notes: The first author was supported by ANR Grant No. ANR-09-JCJC-0102-01.
    The second author was supported by an NSF postdoctoral fellowship.
  • © Copyright 2014 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 27 (2014), 1169-1212
  • MSC (2010): Primary 55N33, 20C20
  • DOI: https://doi.org/10.1090/S0894-0347-2014-00804-3
  • MathSciNet review: 3230821