#### 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 $\ell$-adic cohomology of proper smooth varieties, and more generally by $\ell$-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