Báo cáo khoa học 演講訊息

The overall goal of the lectures is to explain the proofs of two fundamental results about holomorphic curves in Abelian varieties. The first result is the Bloch-Ochiai theorem that the Zariski closure of the image of a holomorphic curve in an Abelian variety is the translate of an Abelian subvariety, and the second result is Siu and Yeung's proof of a conjecture of Lang's that a holomorphic curve omitting an ample divisor in an Abelian variety must be constant. In the process, I will introduce two important techniques: Nevanlinna theory and jet spaces. I will also make comparisons to parallel results in arithmetic geometry and the Kawamata structure theorem. In Lecture 1 I will give an introduction to the lecture series, introduce Nevanlinna theory, and in particular the Lemma on the Logarithmic derivative, and I will introduce the concept of jets. In Lecture 2 I will give Bloch's proof, made rigorous by Ochiai, that the Zariski closure of a holomorphic curve in an Abelian variety must be the translate of an Abelian subvariety. Time permitting, I will also discuss McQuillan's proof to this theorem,which was motivated by Faltings's theorem on rational points on subvarieties of Abelian varieties. In Lecture 3 I will discuss a construction of Siu and Yeung using theta functions to construct a logarithmic jet differential with poles along an ample divisor and explain how the existence of such a jet differential implies that a holomorphic curve omitting an ample divisor in an Abelian variety must be constant. I will conclude with some remarks about more recent results extending the results I will discuss in detail and on limitations that the methods introduced here may have in attacking related problems on general type varieties. A good reference for my lectures is: Chapter 6 of Min Ru, Nevanlinna Theory and Its Relation of Diophantine Approximation, World Scientific, 2001.