A relative bigness inequality and equidistribution theorem over function fields

TL;DR

This paper generalizes Siu's inequality to a relative setting and applies it to establish an equidistribution theorem for small subvarieties over function fields, linking geometric inequalities with arithmetic dynamics.

Contribution

It introduces a relative version of Siu's inequality and demonstrates its application to equidistribution in algebraic dynamics over function fields.

Findings

Generalized Siu's inequality to a relative case

Established an equidistribution theorem for small subvarieties

Connected geometric inequalities with arithmetic dynamics

Abstract

For any line bundle written as a subtraction of two ample line bundles, Siu's inequality gives a criterion on its bigness. We generalize this inequality to a relative case. The arithmetic meaning behind the inequality leads to its application on algebraic dynamic systems, which is the equidistribution theorem of generic and small net of subvarieties over a function field.