On $\psi$-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 $\psi: A\to A$. In this paper, we give an essentially complete classification of all $\psi$-invariant subvarieties of $A$. For example, under some mild assumptions on $(A,\psi)$ 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.