Classification of Lagrangian fibrations

Hyperkahler流形出现在高能理论物理的许多不同领域，例如，作为杨-米尔斯瞬子、磁单极子和希格斯场的参数化空间。在它们的代数几何表示中，超Kahler流形表现为全纯辛流形。全纯辛流形上的纤维称为拉格朗日纤维，因为纤维相对于全纯辛结构必须是拉格朗日纤维。此外，一般纤维必须是阿贝尔变种，拉格朗日纤维可以被视为椭圆K3曲面的高维类似物。希钦系统就是一个著名的非紧凑的例子。在这个项目中，P.I.将研究(紧致)拉格朗日纤维的分类。我们的目标是在某些特殊情况下对拉格朗日纤维进行分类：当纤维是曲线的雅可比时，或者主要是极化的时候，当纤维是局部等平凡的时，在四维(阿贝尔表面作为纤维)。P.I.还将探索拉格朗日纤维的新结构，并解决形变类的有限问题。这个项目将发展一些已知在理论物理中出现的特殊类型的几何，特别是在量子场论和弦理论中。主要的焦点将放在超卡勒几何上，这是一类允许Ricci平坦度量的空间，即在真空中解爱因斯坦方程。P.I.还将研究广义复几何，这是数学家发明的，用来连接复几何和辛几何，IIA和IIB类型的弦理论基于这些几何，并帮助解开弦理论家预测的镜像对称现象。简而言之，这些都是创新的新几何学，主要是为了响应理论物理学家的要求而开发的，但也具有内在的数学兴趣。P.I.将通过组织和参加跨学科会议和研讨会，积极促进数学家和物理学家之间的思想交流，并参与这些主题的研究生培训。

Hyperkahler manifolds arise in many different areas of high energy theoretical physics, for example as spaces parametrizing Yang-Mills instantons, magnetic monopoles, and Higgs fields. In their algebro-geometric manifestation, hyperkahler manifolds appear as holomorphic symplectic manifolds. Fibrations on holomorphic symplectic manifolds are known as Lagrangian fibrations, since the fibers must be Lagrangian with respect to the holomorphic symplectic structure. In addition, the generic fibres must be abelian varieties, and Lagrangian fibrations may be viewed as higher-dimensional analogues of elliptic K3 surfaces. The Hitchin system is a celebrated non-compact example. In this project the P.I. will study the classification of (compact) Lagrangian fibrations. The goals are to classify Lagrangian fibrations in some particular cases: when the fibers are Jacobians of curves or principally polarized, when the fibration is locally isotrivial, in dimension four (abelian surfaces as fibers). The P.I. will also explore new constructions of Lagrangian fibrations, and address the question of finiteness of deformation classes.This project will develop some particular kinds of geometry that are known to arise in theoretical physics, particularly in quantum field theory and string theory. The main focus will be on hyperkahler geometry, a class of spaces admitting Ricci-flat metrics, i.e., solutions to Einstein's equations in a vacuum. The P.I. will also study generalized complex geometry, which was invented by mathematicians to bridge complex and symplectic geometry, on which type IIA and IIB string theory are based, and to help unravel the Mirror Symmetry phenomenon predicted by string theorists. In short, these are all innovative new kinds of geometry developed largely in response to the demands of theoretical physicists, but also of inherent mathematical interest. The P.I. will actively promote the exchange of ideas between mathematicians and physicists by organizing and participating in cross-disciplinary conferences and workshops, and is also involved in the training of graduate students in these topics.