Research

I have been working on Nevanlinna theory and Kobayashi hyperbolicity. Nevanlinna theory, developed by R. Nevanlinna and then generalized in higher-dimensional cases, deals with the distribution of values of meromorphic mappings into a compact complex manifold X with respect to divisors on X. The concept of hyperbolicity due to Kobayashi describes in a precise sense whether a complex manifold contains arbitrarily large transcendental copies of a complex disk. By a remarkable result of Brody, the problem of characterizing the Kobayashi hyperbolicity of compact complex manifold X reduces to discussing the existence or nonexistence of entire curves in X.

Nevanlinna theory consists of two main theorems: the First and the Second. The Second Main Theorem estimates the frequency of the impact of a nondegenerate (nonconstant, algebraically non degenerate, ...) holomorphic curve with the divisors in terms of the characteristic function. This explains why Nevanlinna theory plays an important role in studying the degeneracy problem and Kobayashi hyperbolicity.