Neron models, singularities of normal functions, and Hodge loci

Neron 模型、正态函数奇点和 Hodge 轨迹

• 批准号: 1331641
• 负责人: Christian Schnell
• 金额: $ 7.19万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-05-16 至 2015-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1331641&HistoricalAwards=false
• 关键词: Neron; models; singularities; normal; functions

The PI proposes to apply methods from the theory of mixed Hodge modules to the study of Hodge classes and normal functions, in order to solve several problems about singularities of normal functions and the locus of Hodge classes. The main idea is the construction of natural complex-analytic extension spaces, which make it possible to apply differential-geometric and topological methods to the above problems, and which can also be used to define numerical invariants. To date, the PI has successfully carried out one portion of this, namely the construction of complex-analytic Neron models. He proposes to use his results to describe graph closures of admissible normal functions, to carry out a local study of singularities, and to obtain a classification theorem. The PI also proposes to introduce Poincare bundles of Neron models, both to define numerical invariants for pairs of normal functions and to make progress on the question of the existence of singularities.The proposed work is connected with the program of M. Green and P. Griffiths, and bears on the Hodge conjecture. The general philosophy is to understand the geometry of a complex algebraic variety with the help of differential geometry and topology. The proposed methods can be used, for instance, to study Noether-Lefschetz loci on Calabi-Yau threefolds; this example is interesting because we have very intriguing numerical predictions made by physicists for these loci.

PI提出将混合Hodge模理论的方法应用于Hodge类和正规函数的研究，以解决关于正规函数的奇异性和Hodge类的轨迹的几个问题。其主要思想是构造自然复解析扩张空间，使微分几何和拓扑方法应用于上述问题成为可能，也可用于定义数值不变量。到目前为止，PI已经成功地完成了其中的一部分，即复杂解析Neron模型的构建。他建议用他的结果来描述图封闭的容许正常职能，进行局部研究的奇异性，并获得一个分类定理。PI还建议引入Neron模型的Poincare束，既定义了正规函数对的数值不变量，又在奇点存在性问题上取得了进展。绿色和P.格里菲斯，并对霍奇猜想产生影响。一般的哲学是在微分几何和拓扑学的帮助下理解复代数簇的几何。例如，我们提出的方法可以用来研究卡-丘三重轨道上的诺特-莱夫谢茨轨迹;这个例子很有趣，因为物理学家对这些轨迹做出了非常有趣的数值预测。