Good afternoon,

I'm just curious about this question, because I see that there are a lot of papers which study the value distribution of an entire curve $f\colon \mathbb{C}\to X,$ with X a complex manifold by using Nevanlinna theory. But I don't know the motivation of this research.

My question : Why do we need to study entire curves? Could anyone tell me some references or notes which mention these motivations? And beside the value distribution, what other aspects of entire curves do we want to know/study?

Thanks in advance,

Duc Anh

  • $\begingroup$ So far, I haven't needed it. $\endgroup$ – Fernando Muro Sep 16 '12 at 9:32
  • $\begingroup$ I am always very thrown off when people use the word "need" in this context. Where does this need come from? Who told you that you need to study entire curves? $\endgroup$ – Qiaochu Yuan Sep 18 '12 at 3:27
  • $\begingroup$ As i said in the my question, this question comes from the fact I see that there a lot of papers which deal with the value distribution of an entire curve. So I'm just curious about the motivation of these researches. I don't know well English, so maybe my usage of the word "need" make people misunderstand. $\endgroup$ – Đức Anh Sep 18 '12 at 3:40

A good reference is S. Lang, Introduction to complex hyperbolic geometry. Shortly, I can give the following reasons. Listed in historical order.

  1. There are famous Picard theorems about entire curves whose target is of dimension 1. They are 130 years old and they have a lot of connections with other areas of mathematics. How to generalize them to targets of dimension more than 1 ? There are many conjectures but most of them are still conjectures.

  2. In dimension 1, we have this very useful thing which is called hyperbolic metric. It is one of the main tools in the theory of analytic functions. There is an analog in higher dimensions, called Kobayashi metric. It is also very useful, but the question is: which complex manifolds have it? The answer (for compact manifolds) is given by Brody's Lemma: such metric on a compact manifold exists if and only if there are no non-constant holomorphic curves C to X. That is it reduces the question to the question about holomorphic curves.

  3. There is a deep analogy between the theory of holomorphic curves and Diophantine geometry. That is (very roughly speaking) a manifold over an algebraic number field has finitely many rational points if, as a complex manifold, it is hyperbolic. That is it has no non-constant holomorphic curves.

Nevanlinna theory is one of the main tools of study of holomorphic curves. It was originally developed in the attemts to better understand the nature of Picard's theorems.

This is a very brief answer, and I recommend the book of Lang mentioned above, and his survey articles on holomorphic curves and diophantine geometry. For one-dimensional theory, which is extremelly rich and developed I recommend the books of Nevanlinna himself. They better explain the original motivations then the modern books.

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