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

Sun, Shenghao; Zheng, Weizhe

Algebra Number Theory 10, No. 2, 235-307 (2016).

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

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