Degeneration of the Mukai system

本研究计划的背景是代数几何中自然发生的可积系统的存在。传统上，这样的系统是一个微分方程系统，它有足够的运动常数，原则上可以完全解出方程。几何上，这些运动常数是泊松交换相空间上的函数，其公共水平集具有环面结构。一个经典的例子是摆的运动，它可以用椭圆函数来求解；环面是曲线，或者更准确地说是它的皮卡德变体。另一个更复杂的问题是椭球体上的测地线，它需要超椭圆函数，其中环面是高维的阿贝尔变体。更一般地说，人们可以采用无坐标的观点，考虑由阿贝尔变分构成的辛代数变分，但它们并不容易找到。这种代数变体有两个相当广泛的家族：一个被称为希钦系统，另一个被称为穆凯系统。希钦系统起源于对代数曲线上希格斯束模空间的研究。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曲面上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表明Mukai系统可以看作是Hitchin系统的非线性变形。K3表面已经被深入研究了很多年，特别是被提议的参观者，它们的内部几何形状比躺在它们上面的单个曲线要丰富得多。因此，我们可以期待在Mukai模空间中出现新的特征。最近，希钦系统已成为数论和表示理论等其他数学领域的一个有价值的工具。此外，由于物理学家Witten和Kapustin将其特殊性质与电磁对偶性联系起来的工作，产生了许多新的观点和结果，特别是与朗兰兹纲领有关，朗兰兹纲领是许多数学实体的统一愿景，起源于数论。该建议包括试图理解这些新观点在Mukai系统中的作用：寻找在退化的极限下成为希钦系统中已知结构的新结构。这些结构包括上同调——对空间和自然代表性环的基础拓扑的研究；相干束的派生范畴——一种捕捉空间上复子流形和矢量束之间关系的更精细的方法；以及对用泊松结构的更一般的概念（最初仍然建立在微分方程中）取代K3表面上的辛结构的想法的研究。

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.