Strongly semistable sheaves and the Mordell-Lang conjecture over function fields

TL;DR

This paper offers a new proof of the Mordell-Lang conjecture over function fields in positive characteristic, utilizing semistable sheaves and Frobenius techniques to derive effective bounds and propose related conjectures.

Contribution

It introduces a novel proof approach based on semistable sheaves and Frobenius pull-backs, providing explicit bounds and new conjectures in positive characteristic geometry.

Findings

Effective bounds for isotrivial finite covers

Application of Langer's theorem to Mordell-Lang

Conjectures on cotangent bundle filtrations and Lang conjecture

Abstract

We give a new proof of the Mordell-Lang conjecture in positive characteristic, in the situation where the variety under scrutiny is a smooth subvariety of an abelian variety. Our proof is based on the theory of semistable sheaves in positive characteristic, in particular on Langer's theorem that the Harder-Narasimhan filtration of sheaves becomes strongly semistable after a finite number of iterations of Frobenius pull-backs. The interest of this proof is that it provides simple effective bounds (depending on the degree of the canonical line bundle) for the degree of the isotrivial finite cover whose existence is predicted by the Mordell-Lang conjecture. We also present a conjecture on the Harder-Narasimhan filtration of the cotangent bundle of a smooth projective variety of general type in positive characteristic and a conjectural refinement of the Bombieri-Lang conjecture in positive characteristic.