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，直到一个自然等价，称为双有理等价。两个这样的对象是双有理等价的，如果它们的有理函数空间自然同构。全纯的行为（即，复可微）或代数函数从复数平面与价值在他们是我的调查中心。除了推广Picard定理的经典动机（关于一般类型簇的伪双曲性的Lang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有理数解的有限性的Mordell猜想，如果定义的簇是一般类型的曲线（由G.他因此获得了数学界的最高荣誉--菲尔兹奖。Faltings了两个解决方案的莫德尔猜想，第二个使用的想法Vojta马祖尔之间的平行价值分布理论的全纯曲线，开创了Nevanlinna，和分布的合理或代数点（即解决方案的整数或自由基的整数齐次方程定义的品种）。因此，人们希望，对代数簇中曲线结构的更深入理解，最终将导致解开数论中自然问题与全纯曲线和代数曲线行为之间相似性的奥秘。