Standard conjecture of Hodge type in characteristic p

Determine whether Grothendieck's standard conjecture of Hodge type holds for arbitrary smooth projective varieties over algebraically closed fields of characteristic p, beyond the classes of varieties for which it is currently known.

Background

The paper studies Grothendieck's standard conjecture of Hodge type, which asserts positivity of the relevant intersection form on primitive algebraic cohomology classes. The authors note that, although the conjecture follows from Hodge theory over the complex numbers and is known in several positive-characteristic cases, it remains unresolved in general characteristic p.

The paper proves the conjecture for even-dimensional Hermitian varieties and for varieties finitely covered by them, but does not resolve the conjecture for arbitrary smooth projective varieties in characteristic p.