Holonomic D-modules on abelian varieties

阿贝尔变量同时是代数变量（由多项式方程描述的空间）和阿贝尔群（具有与整数集上的加法遵循相同定律的操作的集合），数学家对它们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，但许多重要问题仍未得到解决。这个项目将采取一种新的方法来解决其中的一些问题，通过研究阿贝尔变体上d模的性质。d -模基本上是偏微分方程的系统，它的抽象形式对研究几何问题很有用。例如，每当有一个从另一个代数变体到一个阿贝尔变体的映射时，就会在这个阿贝尔变体上得到一定数量的d模，这些d模包含有关原始映射的信息。主要的新思想是，这些（和其他）阿贝尔变体上的d模的有趣特性可以在傅里叶分析的帮助下揭示出来，就像信号（如声波）的有趣特性可以通过观察它们的频谱来揭示一样。人们希望对阿贝尔变体上d模的“谱”有一个很好的理解，这将导致关于它们几何的新结果。用更专业的语言来说，该项目的研究目标是给出复阿贝尔变上完整d模的傅立叶-穆凯变换的完整表征。这些变换是具有连接的线束（这是一个超凯勒流形）的模空间上相干束的复合体，并且在许多方面，它们看起来与反常束非常相似。这表明完整d模的Fourier-Mukai变换应该构成一个迄今为止推测的“超kaehler反常束”范畴；该项目将使这一想法变得精确，并测试其一些含义。这个问题也具有实际意义，因为它为解决关于阿贝尔变种和不规则变种的问题提供了一个新的工具，类似于Green和Lazarsfeld的泛型消失定理。第二个目标是使混合Hodge模块理论更广为人知。在最近Clay数学研究所的研讨会上，PI和其他人计划写一本关于混合Hodge模块及其应用的书，试图让这个强大的理论对非专业人士来说也很容易理解。

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.