Degeneration of the Mukai system

向井系统的退化

基本信息

批准号：
EP/H023461/1
负责人：
Nigel James Hitchin
金额：
$ 7.51万
依托单位：
University of Oxford
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2010
资助国家：
英国
起止时间：
2010 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FH023461%2F1
关键词：
Degeneration Mukai system

项目摘要

The background to this research proposal is the existence of naturally occurring integrable systems in algebraic geometry. Traditionally, such a system is a system of differential equations which has enough constants of the motion to allow one in principle to solve completely the equation. Geometrically these constants of the motion are functions on the phase space which Poisson commute and whose common level sets have the structure of a torus. The classical example is the motion of a pendulum which can be solved with elliptic functions; the torus is the curve or more precisely its Picard variety. Another, more complicated one, the geodesics on an ellipsoid, requires hyperelliptic functions, where the torus is a higher-dimensional abelian variety. More generally, one can adopt a coordinate-free viewpoint and consider algebraic varieties which are symplectic and have a fibration by abelian varieties, but they are not so easy to find. There are two quite broad families of such algebraic varieties: one has come to be known as the Hitchin system and the other the Mukai system.The Hitchin system arises from the study of moduli space of Higgs bundles on an algebraic curve. The data consists of a curve and a simple Lie group, and sometimes extra information concerned with marked points on the curve. The Mukai system requires a curve in a K3 surface and a simple Lie group, though most work involves linear groups. Donagi, Ein and Lazarsfeld showed that the Mukai system can be regarded as a nonlinear deformation of the Hitchin system. K3 surfaces have been studied intensely for many years, especially by the proposed visitor, and they have an internal geometry much richer than that of a single curve lying on them. One can therefore expect new features to appear in the Mukai moduli space.The Hitchin system has recently become a valuable tool in other areas of mathematics such as number theory and representation theory. Furthermore, thanks to the work of physicists Witten and Kapustin relating its special properties to electric-magnetic duality, a number of new viewpoints and results have been produced, connecting in particular to the Langlands programme, a unifying vision of many mathematical entities which originates in number theory. The proposal consists of attempting to understand the role of these new points of view in the Mukai system: looking for new structures which in the limit of the degeneration become the known ones on the Hitchin system. The structures include cohomology -- the study of the underlying topology of the spaces and natural representative cycles; the derived category of coherent sheaves -- a more refined way of capturing the relations between complex submanifolds and vector bundles over the space; and an investigation into the idea of replacing the symplectic structure on the K3 surface by the more general notion (still originally founded in differential equations) of a Poisson structure.

该研究建议的背景是在代数几何形状中存在自然存在的整合系统。传统上，这样的系统是一个微分方程的系统，具有足够的运动常数，可以原则上允许一个人完全求解方程。从几何上讲，这些运动的常数在泊松通勤的相空间上起作用，其公共水平集具有圆环的结构。经典的例子是可以用椭圆函数求解的摆的运动。圆环是曲线或更精确的Picard品种。另一个更复杂的是，椭圆形上的大地测量需要过度纤维化功能，其中圆环是高维的阿贝尔人品种。更一般而言，人们可以采用无坐标的观点，并考虑具有符合性并具有Abelian品种纤维化的代数品种，但并不容易找到。这种代数品种有两个相当广泛的家族：一个被称为Hitchin系统，另一个被称为Mukai系统。Hitchin系统源于对Higgs捆绑的模量在代数曲线上捆绑的空间。数据由曲线和简单的谎言组组成，有时还包括有关曲线上明显点的额外信息。 Mukai系统需要在K3表面和简单的谎言组中曲线，尽管大多数工作都涉及线性组。多纳吉，EIN和Lazarsfeld表明，Mukai系统可以视为Hitchin系统的非线性变形。多年来，已经对K3表面进行了深入研究，尤其是由拟议的访客进行了研究，并且它们的内部几何形状比单个曲线的几何形状丰富。因此，人们可以期望新功能出现在Mukai Moduli空间中。Hitchin系统最近已成为数学其他领域（例如数字理论和表示理论）的宝贵工具。此外，由于物理学家Witten和Kapustin的工作将其特殊属性与电磁二元性联系起来，已经产生了许多新的观点和结果，特别是与Langlands计划建立了联系，这是许多数学实体的统一愿景，这些数学实体以数量理论来源。该提案包括试图了解这些新观点在Mukai系统中的作用：寻找新结构，这些结构在堕落的极限中成为Hitchin系统上已知的结构。这些结构包括共同体 - 对空间的潜在拓扑结构和自然代表周期的研究；连贯的滑轮的派生类别 - 一种更精致的方式，是捕获整个空间上复杂的子手机和向量束之间的关系；以及对泊松结构的更通用的概念（仍然最初建立在微分方程中）代替K3表面上的符号结构的想法的研究。