Neron models, singularities of normal functions, and Hodge loci

PI提出将混合Hodge模理论的方法应用到Hodge类和正规函数的研究中，以解决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，它使微分几何和拓扑方法应用于上述问题成为可能，并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，即构建复杂分析的Nelon模型。他建议用他的结果来描述可容许正规函数的图闭包，进行奇点的局部研究，并得到一个分类定理。PI还建议引入Poincare丛，以定义正态函数对的数值不变量，并在奇点存在的问题上取得进展。其基本原理是借助微分几何和拓扑学来理解复杂代数簇的几何。例如，所提出的方法可以用来研究Calabi-Yau三重上的Noether-Lefschetz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。

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.