Holonomic D-modules on abelian varieties

阿贝尔簇同时是代数簇（由多项式方程描述的空间）和阿贝尔群（具有与整数集合上的加法相同的运算法则的集合），而数学家对它们感兴趣的是这两种结构之间的相互作用。自世纪以来，特别是在代数几何领域，人们对它们进行了深入的研究，但仍有许多重要问题没有解决。本项目将通过研究阿贝尔簇上D-模的性质，对其中的一些问题采取新的方法。D-模基本上是偏微分方程的系统，其抽象形式对研究几何问题很有用。例如，当有一个从另一个代数簇到一个阿贝尔簇的映射时，我们得到阿贝尔簇上一定数量的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）D-模在交换簇上的有趣性质可以借助傅立叶分析来揭示，就像信号（如声波）的有趣性质可以通过观察它们的频谱来揭示一样。希望对阿贝尔簇上D-模的“谱”有一个很好的理解，这将导致关于其几何的新结果。用更专业的语言来说，该项目的研究目标是给出复阿贝尔簇上完整D-模的Fourier-Mukai变换的完整刻画。这些变换是具有联络的线丛的模空间（这是一个超凯勒流形）上的相干层的复体，在许多方面，它们看起来非常类似于反常层。这表明完整D-模的傅里叶-向井变换应该构成一个迄今为止的“超凯勒反常层”的理论范畴;该项目将使这一想法精确化，并测试它的一些含义。这个问题也是有实际意义的，因为它提供了一个新的工具来解决关于交换簇和不规则簇的问题，类似于绿色和Lazarsfeld的通有消失定理。第二个目标是使混合霍奇模的理论更加广为人知。在最近的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及其应用的书，试图让非专家也能接触到这个强大的理论。

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.