Partial Heights and the Geometric Bombieri-Lang Conjecture

We prove the geometric Bombieri-Lang conjecture for projective varieties which have finite morphisms to abelian varieties over function fields of characteristic 0. Our proof is complex analytic, which applies the classical Brody lemma to construct entire curves on complex varieties. Our key ingredients includes a new notion of partial height and its non-degeneracy in a suitable sense. The non-degeneracy is required in the application of the Brody lemma.Comment: 61 page

Effective bound of linear series on arithmetic surfaces

We prove an effective upper bound on the number of effective sections of a hermitian line bundle over an arithmetic surface. It is an effective version of the arithmetic Hilbert--Samuel formula in the nef case. As a consequence, we obtain effective lower bounds on the Faltings height and on the self-intersection of the canonical bundle in terms of the number of singular points on fibers of the arithmetic surface

On Volumes of Arithmetic Line Bundles

We show an arithmetic generalization of the recent work of Lazarsfeld-Mustata which uses Okounkov bodies to study linear series of line bundles. As applications, we derive a log-concavity inequality on volumes of arithmetic line bundles and an arithmetic Fujita approximation theorem for big line bundles.Comment: 21 page

The Geometric Bombieri-Lang Conjecture for Ramified Covers of Abelian Varieties

In this paper, we prove the geometric Bombieri-Lang conjecture for projective varieties which have finite morphisms to abelian varieties of trivial traces over function fields of characteristic 0. The proof is based on the idea of constructing entire curves in the pre-sequel "Partial heights, entire curves, and the geometric Bombieri-Lang conjecture." A new ingredient is an explicit description of the entire curves in terms of Lie algebras of abelian varieties.Comment: The original paper arXiv:2305.14789v1 is split into two papers: arXiv:2305.14789v2 and the current pape