演講訊息 TALKS

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.