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2016 Parity and symmetry in intersection and ordinary cohomology
Shenghao Sun, Weizhe Zheng
Algebra Number Theory 10(2): 235-307 (2016). DOI: 10.2140/ant.2016.10.235


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.


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Shenghao Sun. Weizhe Zheng. "Parity and symmetry in intersection and ordinary cohomology." Algebra Number Theory 10 (2) 235 - 307, 2016.


Received: 26 August 2014; Revised: 2 October 2015; Accepted: 31 December 2015; Published: 2016
First available in Project Euclid: 16 November 2017

zbMATH: 1375.14071
MathSciNet: MR3477743
Digital Object Identifier: 10.2140/ant.2016.10.235

Primary: 14F20
Secondary: 11E81 , 14F43 , 14G15 , 14G25

Keywords: $\ell$-adic cohomology , alteration , alternating form , Decomposition theorem , Deligne–Mumford stack , Galois representation , Grothendieck–Witt group , horizontal complex , intersection cohomology , pure perverse sheaf , symmetric form

Rights: Copyright © 2016 Mathematical Sciences Publishers


Vol.10 • No. 2 • 2016
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