Asymptotic Hodge Theory and Instantons

摘要奖：DMS-1005761首席研究员：马克·A·斯特恩PI提出的研究集中在与联络的存在和几何有关的几个问题上，最小化特殊完整流形上的向量丛和光滑射影簇上的联络的Yang-Mills和相关的能量泛函。在第一个项目中，PI将与Bianca Santoro合作，研究由曲率张量的(0，2)分量的L2范数给出的能量E‘的临界点和抛物线流动。在许多De Rham上同调都是(p，p)型的特殊几何中，他们将试图证明E‘的稳定临界点的联络的全纯性。他们还将研究阻碍E‘的梯度流解的长期存在的障碍。这个项目的主要目标是提高我们对向量丛何时允许全纯结构的理解。在与Benoit Charbonneau合作的一个相关项目中，PI将使用几何量子化的技术来研究光滑射影簇上向量丛的同构类上的精化Hodge结构。他们将研究由极化线丛的k次方张量的丛F上的广义扩散算子在核上的投影算子的大k渐近性.他们利用这些渐近性定义了丛上的精化Hodge滤子，并试图进一步利用这些渐近性来证明从丛到De Rham上同调的Chern特征标映射(配备其通常的Hodge结构)遵守这些新的Hodge滤子.在这项研究中出现了唐纳森平衡度量条件的关联类比。PI将寻求与吉赛克稳定性相应的类比。在与萨夫迪普·塞西的合作中，PI将研究RXT^3上瞬子的模空间，在T^3上的平坦连接之间进行内插，具有不等的陈西蒙斯不变量。强、弱和电磁力的物理学基于规范理论，即对矢丛上的微积分的研究。这些规范理论的主要对象是联系，它可以被视为定义差异的规则。在这些物理理论中，指导动力学的主要能量是杨-米尔斯能量，而主导物理的场是那些最小化杨-米尔斯能量的场。在大于4的维度中，这些最优连接的存在和结构很难理解。在4维空间中，最优连接一直是回答几何和拓扑问题的有力工具。在更高的维度中，它们可能为解决微分几何和代数几何中的问题提供极其重要的技术。特别是，它们可以给出将涉及复杂偏微分方程解的问题转化为极其简单的代数问题的技术。由于目前的量子引力理论都需要理解维度大于4的物理，所以对最佳连接的物理和几何研究之间存在着很强的相互作用。事实上，这个项目的前奏源于物理学家向PI提出的问题；这些问题的答案随后导致了几何上的新结果。

AbstractAward: DMS-1005761Principal Investigator: Mark A. SternThe PI's proposed research focuses on several problems related to the existence and geometry of connections minimizing the Yang-Mills and related energy functionals of connections on vector bundles over manifolds of special holonomy and on smooth projective varieties. In the first project the PI, in joint work with Bianca Santoro, will study critical points and parabolic flows of the energy, E', given by the L2 norm of the (0,2)component of the curvature tensor. In many special geometries where the de Rham cohomology is all of type (p,p), they will attempt to prove holomorphicity of connections that are stable critical points of E'. They will also study obstructions to the long time existence of solutions to the gradient flow for E'. The primary goal of this project is to improve our understanding of when vector bundles admit holomorphic structures. In a related project, joint with Benoit Charbonneau, the PI will use techniques from geometric quantization to study refined Hodge structures on isomorphism classes of vector bundles on smooth projective varieties. They will study the large k asymptotics of the projection operator onto the kernel of a generalized Diracoperator associated to a bundle F tensored by the kth power of a polarizing line bundle.They have used these asymptotics to define a refined Hodge filtration on bundles and will attempt to exploit the asymptotics further to prove that the Chern character mapping from bundles to de Rham cohomology (equipped with its usual Hodge structure) respects these new Hodge filtrations. Connection analogs of Donaldson's balanced metric conditions arise in this study. The PI will seek corresponding analogs of Gieseker stability. In joint work with Savdeep Sethi, the PI will study the moduli space of instantons on RxT^3, interpolating between flat connections on T^3, with unequal Chern Simons invariants.The physics of the strong, weak, and electromagnetic forces is based on gauge theory, the study of differential calculus on vector bundles. The primary objects in these gauge theories are connections, which may be viewed as rules for defining differentiation. The primary energy guiding the dynamics in these physical theories is the Yang-Mills energy, and the fields dominating the physics are those that minimize the Yang-Mills energy. In dimensions greater than 4, the existence and structure of these optimal connections are poorly understood. In 4-dimensions, optimal connections have been a powerful tool for answering questions in geometry and topology. In higher dimensions they can potentially provide extremely important techniques for attacking questions in differential and algebraic geometry. In particular, they may give techniques for transforming questions involving the solutions of complex partial differential equations to dramatically simpler algebraic problems. As the current theories of quantum gravity all require an understanding of physics in dimensions greater than 4, there is strong interplay between the physical and geometric study of optimal connections. In fact, the antecedents to this project arose from questions posed to the PI by physicists; the answer to these questions then led to new results in geometry.