Holonomic D-modules on abelian varieties

阿贝尔簇同时也是代数簇(用多项式方程描述的空间)和阿贝尔群(其运算遵循与整数集上的加法相同的规律的集合)，而数学家感兴趣的是这两种结构之间的相互作用。自19世纪以来，人们对它们进行了深入的研究，特别是在代数几何领域，但许多重要的问题仍然没有得到解决。这个项目将通过研究交换簇上的D-模的性质，对其中一些问题采取一种新的方法。D-模基本上是偏微分方程组，它是一种抽象形式，对研究几何问题很有用。例如，当一个人有一个从另一个代数簇到一个交换簇的映射时，就会得到该交换簇上的一定数量的D-模，它们包含关于原始映射的信息。主要的新思想是，这些(和其他)阿贝尔变种上的D-模的有趣性质可以通过傅立叶分析来揭示，就像信号(例如声波)的有趣性质可以通过观察它们的频谱来揭示一样。用更专业的语言来描述复杂交换簇上完整D-模的傅里叶-Mukai变换是本课题的研究目标。这些变换是具有连接的线丛(这是一个超Kaehler流形)的模空间上的相干层的复形，在许多方面，它们看起来与倒立的层非常相似。这表明，完整D-模的傅里叶-穆凯变换应该构成一个迄今尚未被猜想的范畴--“Hyperkaehler Perverse Sheet”；该项目将使这一想法变得精确，并测试它的一些含义。这个问题也很有实际意义，因为它为解决交换变元和不规则变元的问题提供了一种新的工具，类似于Green和Lazarsfeld的一般消失定理。第二个目标是使混合霍奇模的理论更广为人知。在最近克莱数学研究所研讨会的基础上，PI和其他人计划写一本关于混合霍奇模块及其应用的书，试图让非专家理解这一强大的理论。

Abelian varieties are at the same time algebraic varieties (spaces described by polynomial equations) and abelian groups (sets with an operation that obeys the same laws as addition on the set of whole numbers), and what makes them interesting to a mathematician is the interplay between those two structures. They have been intensively studied since the 19th century, especially in the field of algebraic geometry, but many important questions remain unsolved. This project will take a new approach to some of those questions, by investigating the properties of D-modules on abelian varieties. D-modules are basically systems of partial differential equations, in an abstract form that is useful for studying geometric questions. For example, whenever one has a mapping from another algebraic variety to an abelian variety, one gets a certain number of D-modules on the abelian variety that contain information about the original mapping. The main new idea is that interesting properties of these (and other) D-modules on abelian varieties can be revealed with the help of Fourier analysis, in the same way that interesting properties of signals (such as sound waves) can be revealed by looking at their frequency spectrum. The hope is that a good understanding of the "spectrum" of D-modules on abelian varieties will lead to new results about their geometry.In more technical language, the research objective of the project is to give a complete characterization of Fourier-Mukai transforms of holonomic D-modules on complex abelian varieties. These transforms are complexes of coherent sheaves on the moduli space of line bundles with connection (which is a hyperkaehler manifold), and in many ways, they look remarkably similar to perverse sheaves. This suggests that Fourier-Mukai transforms of holonomic D-modules should make up an as-yet conjectural category of "hyperkaehler perverse sheaves"; the project will make this idea precise and test some of its implications. This question is also of practical interest, because it provides a new tool for solving problems about abelian varieties and irregular varieties, similar to the generic vanishing theorem of Green and Lazarsfeld. A second objective is to make the theory of mixed Hodge modules more widely known. Building on a recent Clay Mathematics Institute workshop, the PI and others plan to write a book about mixed Hodge modules and their applications that will try to make this powerful theory accessible to non-experts.