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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.

这项研究建议的背景是存在自然发生的代数几何可积系统。传统上，这样的系统是一个微分方程组，它有足够的运动常数，原则上可以完全求解方程。几何上这些常数的运动是函数的相空间的泊松交换和其共同的水平集有一个环面的结构。经典的例子是摆的运动，可以用椭圆函数求解;环面是曲线，或者更精确地说是它的皮卡品种。另一个，更复杂的，椭球上的测地线，需要超椭圆函数，其中环面是一个高维阿贝尔变种。更一般地说，人们可以采用一个坐标自由的观点，并考虑代数簇是辛的，并有一个纤维化的阿贝尔簇，但他们不那么容易找到。有两个相当广泛的家庭这样的代数簇：一个已被称为希钦系统和其他Mukai系统。希钦系统产生于研究模空间的希格斯丛的代数曲线。数据由一条曲线和一个简单的李群组成，有时还包括与曲线上的标记点有关的额外信息。Mukai系统需要K3曲面中的一条曲线和一个简单的李群，尽管大多数工作涉及线性群。Donagi，Ein和Lazarsfeld证明了Mukai系统可以看作是Hitchin系统的非线性变形。K3曲面已经被深入研究了很多年，特别是被提议的访问者，它们的内部几何形状比躺在它们上面的单个曲线丰富得多。因此，人们可以期待新的功能出现在Mukai模空间。希钦系统最近已成为一个有价值的工具，在其他领域的数学，如数论和表示论。此外，由于物理学家维滕和卡普斯廷将其特殊性质与电磁对偶性联系起来，产生了许多新的观点和结果，特别是与朗兰兹纲领联系在一起，朗兰兹纲领是起源于数论的许多数学实体的统一观点。该建议包括试图理解这些新的观点在向井系统中的作用：寻找新的结构，在退化的极限成为已知的希钦系统。结构包括上同调--研究空间和自然代表圈的基本拓扑;相干层的导出范畴--捕捉空间上复子流形和向量丛之间关系的一种更精细的方法;并对用更一般的概念代替K3曲面上的辛结构的思想进行了研究（仍然最初建立在微分方程）的泊松结构。