Slope Boundedness and Equidistribution Theorem

In this article, we prove the boundedness of minimal slopes of adelic line bundles over function fields of characteristic $0$. This can be applied to prove the equidistribution of generic and small points with respect to a big and semipositive adelic line bundle. Our methods can be applied to the finite places of number fields as well. We also show the continuity of χ-volumes over function fields.