## Parity and symmetry in intersection and ordinary cohomology.(English)Zbl 1375.14071

Summary: 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.

### MSC:

 14F20 Étale and other Grothendieck topologies and (co)homologies 14G15 Finite ground fields in algebraic geometry 14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) 14G25 Global ground fields in algebraic geometry 11E81 Algebraic theory of quadratic forms; Witt groups and rings
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