Hyperbolicity and classification theory in complex algebraic geometry

我的工作涉及代数族的结构和分类，代数族是由齐次多项式定义的对象，直到自然等价，称为二元等价。如果两个这样的对象的有理函数空间自然同构，则它们在出生上是等价的。在复数平面上具有值的全纯(即，复可微)或代数函数的行为是我研究的中心。除了推广Picard定理的经典动机(Lang关于一般类型变种的伪双曲性的猜想)之外，还有来自数论的强烈动机，例如Mordell猜想，如果定义的变元是一般类型的曲线，则关于一组多项式的解关于有理数的有限(由G.Faltings解决，他因此获得了数学中的最高荣誉，菲尔兹奖)。Faltings对Mordell猜想给出了两种解决方案，第二种使用Vojta-Mazur的思想，即由Nevanlinna开创的全纯曲线的值分布理论与有理或代数点的分布之间的平行关系(即通过整数或定义种类的齐次方程的整数根来求解)。因此，人们希望对代数变种中曲线结构的更深层次的理解将最终揭开数论中自然问题与关于全纯曲线和代数曲线行为的问题之间相似之处的奥秘。

My work involves the structure and classification of algebraic varieties, which are objects defined by homogeneous polynomials, up to a natural equivalence, called birational equivalence. Two such objects are birationally equivalent if their spaces of rational functions, are naturally isomorphic. The behavior of holomorphic (i.e., complex differentiable) or algebraic functions from the complex number plane with values in them lies at the center of my investigation. Besides the classical motivation of generalizing Picard's theorems (Lang's conjecture on the pseudo-hyperbolicity of varieties of general type), there is also strong motivation from number theory, such as the Mordell conjecture on the finiteness of solutions to a set of polynomials in terms of rational numbers if the variety defined is a curve of general type (solved by G. Faltings, for which he obtained the highest honor in Mathematics, the Fields Medal). Faltings gave two solutions to the Mordell conjecture, the second using an idea of Vojta-Mazur on the parallel between the value distribution theory of holomorphic curves, Pioneered by Nevanlinna, and the distribution of rational or algebraic points (i.e. solutions by integers or by radicals of integers of the homogeneous equations defining the variety). It is therefore hoped that a deeper understanding of the structure of curves in an algebraic variety would eventually lead to unlocking the mysteries of the similarities seen between natural questions in number theory and those concerning the behavior of holomorphic and algebraic curves.