Hyperbolicity and classification theory in complex algebraic geometry

复代数几何中的双曲性和分类理论

• 批准号: 170276-2010
• 负责人: Lu, Steven
• 金额: $ 1.09万
• 依托单位: Université du Québec à Montréal
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2013
• 资助国家: 加拿大
• 起止时间: 2013-01-01 至 2014-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=522780
• 关键词: Hyperbolicity; classification; theory; complex; algebraic

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.

我的工作涉及代数簇的结构和分类，代数簇是由齐次多项式定义的对象，直到自然等价，称为双有理等价。如果两个这样的对象的有理函数空间自然同构，则它们是双有理等价的。来自复数平面的全纯（即复数可微）或代数函数及其值的行为是我研究的中心。除了推广皮卡德定理的经典动机（朗关于一般类型簇的伪双曲性的猜想）之外，还有来自数论的强烈动机，例如关于一组多项式的解的有限性的莫德尔猜想有理数，如果定义的变量是一般类型的曲线（由 G. Faltings 解决，他因此获得了数学最高荣誉菲尔兹奖）。法尔廷斯对莫德尔猜想给出了两个解决方案，第二个解决方案使用了 Vojta-Mazur 的思想，该思想由 Nevanlinna 首创，与有理点或代数点的分布（即整数或根式的解）之间的平行全纯曲线值分布理论相关。定义品种的齐次方程的整数）。因此，希望对代数簇中曲线结构的更深入理解最终能够解开数论中的自然问题与有关全纯和代数曲线行为的问题之间的相似性之谜。