Generalized Kuga–Satake theory and rigid local systems, II : rigid Hecke eigensheaves

Patrikis, Stefan

doi:10.2140/ant.2016.10.1477

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Generalized Kuga–Satake theory and rigid local systems, II: rigid Hecke eigensheaves

Stefan Patrikis

Vol. 10 (2016), No. 7, 1477–1526

DOI: 10.2140/ant.2016.10.1477

Abstract

We use rigid Hecke eigensheaves, building on Yun’s work on the construction of motives with exceptional Galois groups, to produce the first robust examples of “generalized Kuga–Satake theory” outside the Tannakian category of motives generated by abelian varieties. To strengthen our description of the “motivic” nature of Kuga–Satake lifts, we digress to establish a result that should be of independent interest: for any quasiprojective variety over a (finitely generated) characteristic-zero field, the associated graded of the weight filtration on its intersection cohomology arises from a motivated motive in the sense of André, and in particular from a classical homological motive if one assumes the standard conjectures. This extends work of de Cataldo and Migliorini.

Keywords

Galois representations, rigid local systems, Kuga–Satake construction, geometric Langlands

Mathematical Subject Classification 2010

Primary: 14C15

Secondary: 11F80, 14D24

Milestones

Received: 6 June 2015

Revised: 6 January 2016

Accepted: 11 June 2016

Published: 27 September 2016

Authors

Stefan Patrikis
Department of Mathematics University of Utah 155 S 1400 E Salt Lake City, UT 84112 United States

Vol. 10, No. 7, 2016