Complex Analysis and Geometry

复分析与几何所提出的研究涉及复分析与代数或辛几何相结合的几个领域。概述的主要项目旨在发展一个Delzant理论来分类紧流形上的完全可积系统，其作用变量具有实代数型奇点。这些意味着包括由球形品种（Gelfand-Cetlin例子）产生的系统和至少由箭状品种产生的系统。关键工具是波尔-索默菲尔德水平集及其与哈密顿动力学中矩映射几何的相互作用。其他要处理的主题包括外蒙日-安培解的正则性及其与多势理论的关系；复流形无穷远处的渐近几何及其上的重整陈氏类关于由K3曲面导出的高维复流形上的特殊Hodge类的Hodge猜想的一些例子，以及某些Grauert管的代数化。从哲学上讲，第一个项目是基于量子力学开始时的一个想法：机械系统的玻尔-索默菲尔德水平，经典机械系统的一组配置，将对系统的量子力学理解做出重大贡献，例如，原子的光谱线。后来，人们用波函数对力学进行了双重描述，并将两者联系起来，成为物理学和数学中伟大见解的来源。我们怀疑我们已经发现并试图证明的是，玻尔-索默菲尔德对应关系是经典力学系统中某些分类问题的基础：有多少个可积系统？玻尔-索默菲尔德能级与机械系统底层空间上的经典轨迹直接相关，而相应的波函数允许人们读出系统的代数几何。提出的主要问题是使这种“对应”变得严格，以至于系统可以通过玻尔-索默菲尔德轨迹和一些全局代数几何数据之间的组合关系进行分类。

Complex Analysis and GeometryThe research proposed deals with several areas on the interface of complex analysis and algebraic or symplectic geometry. The main project outlined seeks to develop a Delzant theory to classify completely integrable systems on compact manifolds with real algebraic type singularities of their action variables. These are meant to include the systems arising from spherical varieties (Gelfand-Cetlin examples) and those from quiver varieties at a minimum. The key tool would be Bohr-Sommerfeld level sets and their interplay with the geometry of the moment map in Hamiltonian dynamics. Other topics to be treated include regularity of exterior Monge-Ampere solutions and their relation to pluripotential theory; asymptotic geometry at infinity of complex manifolds, and renormalized Chern classes on them; some cases of the Hodge conjecture related to special Hodge-classes on higher dimensional complex manifolds derived from K3 surfaces, and the algebraicization of certain Grauert tubes.Philosophically, the first project is based on an idea going back to the beginning of quantum mechanics: the Bohr-Sommerfeld levels of a mechanical system, the set of configurations of a classical mechanical system which would contribute significantly to a quantum mechanical understanding of the system, as in, e.g., spectral lines of atoms. There was a later dual description of mechanics in terms of wave functions, and relating the two has been a source of great insight in physics and mathematics. What we suspect we have found, and are trying to prove, is that this Bohr-Sommerfeld correspondence underlies certain classification questions in classical mechanicalsystems: how many integrable systems are there? The Bohr-Sommerfeld levels are related directly to classical loci on the underlying space of the mechanical system, while the corresponding wave functions allow one to read off the algebraic geometry of the system. The main problem proposed is to make this "correspondence" rigorous, to the point where the systems can be classified by the combinatorial relations among the Bohr-Sommerfeld loci, and some global algebraic geometric data.