306 research outputs found
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
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