Vol. 10, No. 2, 2016

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Parity and symmetry in intersection and ordinary cohomology

Shenghao Sun and Weizhe Zheng

Vol. 10 (2016), No. 2, 235–307
Abstract

We show that the Galois representations provided by -adic cohomology of proper smooth varieties, and more generally by -adic intersection cohomology of proper varieties, over any field, are orthogonal or symplectic according to the degree. We deduce this from a preservation result of orthogonal and symplectic pure perverse sheaves by proper direct image. We show, moreover, that the subgroup of the Grothendieck group generated by orthogonal pure perverse sheaves of even weights and symplectic pure perverse sheaves of odd weights are preserved by Grothendieck’s six operations. Over a finite field, we deduce parity and symmetry results for Jordan blocks appearing in the Frobenius action on intersection cohomology of proper varieties, and virtual parity results for the Frobenius action on ordinary cohomology of arbitrary varieties.

To the memory of Torsten Ekedahl

Keywords
$\ell$-adic cohomology, intersection cohomology, Galois representation, symmetric form, alternating form, pure perverse sheaf, decomposition theorem, horizontal complex, alteration, Deligne–Mumford stack, Grothendieck–Witt group
Mathematical Subject Classification 2010
Primary: 14F20
Secondary: 14G15, 14F43, 14G25, 11E81
Milestones
Received: 26 August 2014
Revised: 2 October 2015
Accepted: 31 December 2015
Published: 16 March 2016
Authors
Shenghao Sun
Yau Mathematical Sciences Center
Tsinghua University
Jinchunyuan West Building
Beijing, 100084
China
Weizhe Zheng
Morningside Center of Mathematics
Academy of Mathematics and Systems Science, Chinese Academy of Sciences
Zhongguancun Donglu 55
Beijing, 100190
China