On ψ/invariant subvarieties of semiabelian varieties and the Manin-Mumford conjecture

Let A be a semiabelian variety over an algebraically closed field of arbitrary characteristic, endowed with a finite morphism psi : A --> A. In this paper, we give an essentially complete classification of all psi-invariant subvarieties of A. For example, under some mild assumptions on (A, 0) we prove that every psi-invariant subvariety is a finite union of translates of semiabelian subvarieties. This result is then used to prove the Manin-Mumford conjecture in arbitrary characteristic and in full generality. Previously, it had been known only for the group of torsion points of order prime to the characteristic of K. The proofs involve only algebraic geometry, though scheme theory and some arithmetic arguments cannot be avoided.