Coherent cohomology groups/positive characteristic/subgroups/description

Let be a projective variety over a field of positive characteristic , a coherent sheaf on and consider

.

Do there exist any interesting -subspaces of ?

  • Classes which are annihilated by some Frobenius power.
  • Classes which are annihilated under some finite extension .
  • Classes with the property: there exists a , , such that
for all .
  • For and a vector bundles one can also look for properties of the geometric torsor defined by a class . The class defines an extension

and hence projective bundles

and the corresponding torsor

.

Are there subspaces related to properties of ?