Cycles, Characters and Pluripotential Theory in Calibrated Geometry

校准几何中的循环、特征和多能理论

• 批准号: 0705467
• 负责人: H. Blaine Lawson
• 金额: $ 43.9万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-06-01 至 2011-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0705467&HistoricalAwards=false
• 关键词: Cycles; Characters; Pluripotential; Theory; Calibrated

Part IThis project is concerned with the study of cycles and their boundaries, forms, and generalized plurisubharmonic functions. The proposal has several interrelated parts. The first concerns the groups of algebraic cycles and cocycles on a projective variety X. The aim is to relate these groups to the global structure of X. The investigator has, with others, established a theory of homology type for algebraic varieties based on the homotopy groups of cycles spaces. This theory will be used to study concrete questions about algebraic spaces. Implications for real algebraic geometry will be explored. Striking connections to universal constructions in topology which emerged in prior research will also be investigated. The second part of the proposal concerns cycles which bound holomorphic chains in projective manifolds. In particular, characterizations in terms of projective linking numbers and quasi-plurisubharmonic functions will be sought. This will entail a deep analysis of the structure of projective hulls, a concept analogous to polynomial hulls, which has been introduced by the investigator and is ofindependent interest. Projective hulls are related to approximation theory, pluripotential theory, and the spectrum of Banach graded algebras. The third topic, a major part of the proposal, concerns the broad development of a pluripotential theory in calibrated and other geometries. The notions of plurisubharmonic function, pseudo-convex domain, capacity, and solutions to the Dirichlet problem for generalized Monge-Amp\`ere-type equations will be studied in a very general setting. This project, already underway, should have an impact in calibrated geometry, which in turn plays an important role in M-theory in modern physics. There should also be applications to symplectic geometry and to p-convexity in riemannian geometry. The forth part of the proposal concerns sparks and spark complexes. These objects mediate betweeen cycles and smooth data, and give a concrete presentation of differential characters and their generalizations. In the complex category this involves an analytic study of Deligne cohomology and relates to arithmetic Chow groups. It yields invariants for bundles and foliations, and retrieves the classical Abel-Jacobi mappings. This project will also be concerned with student development, including an undergraduate educational effort aimed at fostering mathematical independence and developing interactive environments. Part II A concept of central importance in geometry is that of a ``cycle''. In algebraic geometry a cycle corresponds to the simultaneous solution of a system of polynomial equations. In differential geometry they arise in many ways: as the large scale solutions of certain differential equations, and as the level sets and singularity sets of differentiable mappings. Curves and surfaces in space are simple examples. Cycles with a particular geometry also play a fundamental role in modern physical theoriesThis proposal is concerned with the study of cycles across this broad spectrum. In the algebraic setting, cycles have been related to fundamental large-scale geometry of their surrounding space. This discovery has revealed surprizing and important relationships between spaces of algebraic cycles and fundamental constructions in algebraic topology and has led to new insights in both fields. This work will be continued.Another area of investigation concerns cycles which form the boundary of subsets with special geometric structure. They represent non-linear versions of classical boundary value problems in analysis. Such questions arise in many contexts. The proposer has formulated conjectures relating important classes of such cycles to questions in approximation theory and Banach algebras. Successful resolution will establish a series of new results in complex geometry and should lead to significant new insights in several other fields of mathematics.A third part of the proposal aims at extending classical pluripotential theory to very general geometric settings. These include calibrated geometries, symplectic and Lagrangian geometries, and much more.An uncanny amount of the classical theory has already been shown to hold in this general context. Solutions to the Dirichlet problem for associated Monge-Ampere type equations will be sought in this setting. The study is, in a certain strict sense, dual to the study of the special cycles appearing in these geometries. It should apply to Special Lagrangian cycles in Calabi-Yau manifolds, and associative and Cayley cycles in G(2) and Spin(7) spaces. These latter subjects relate to gauge field theory and gravity in PhysicsA forth domain of investigation concerns a mathematical apparatus developed by the proposer and R. Harvey to detect subtle relationships between cycles and the global structure of the space they live in. This apparatus encompasses some of the most effective tools historically developed for this purpose, and it is much more general. Further development of this theory and its applications will be pursued. This project will also be concerned with graduate student development.Students will be part of the research team. There will also be an undergraduate educational effort aimed at fostering mathematical independence and developing interactive environments.

第一部分本项目主要研究圈及其边界、形式和广义多重次调和函数。该提案有几个相互关联的部分。第一个是关于射影簇X上的代数圈群和上圈群。目的是将这些群体与全球结构联系起来， X. 研究者与他人建立了基于圈空间同伦群的代数簇的同调型理论。这个理论将被用来研究有关代数空间的具体问题。 将探索对真实的代数几何的影响。在拓扑结构中出现在先前的研究中的普遍结构的惊人的连接也将被调查。 第二部分的建议关注的周期约束全纯链在射影流形。特别是，将寻求在投影连接数和拟pluisubharmonic函数的特征。这将需要深入分析的结构的射影壳，一个概念类似于多项式壳，这已经介绍了由调查员和独立的兴趣。射影壳与逼近理论、多能理论和Banach分次代数的谱有关。 第三个主题，该提案的一个主要部分，涉及校准和其他几何形状的多能理论的广泛发展。本文将在一个非常一般的背景下研究广义Monge-Schiller型方程Dirichlet问题的多重次调和函数、伪凸域、容度和解的概念。 这个项目已经在进行中，应该会对校准几何产生影响，这反过来又在现代物理学的M理论中发挥重要作用。 也应该有应用辛几何和p-凸黎曼几何。建议的第四部分是关于火花和火花复合体。这些对象在循环和光滑数据之间起中介作用，并给出了微分特征及其推广的具体表示。在复杂的范畴，这涉及分析研究德利涅上同调，并涉及算术周群。它产生不变量的丛和叶状，并检索经典的Abel-Jacobi映射。 该项目还将关注学生发展，包括旨在培养数学独立性和开发互动环境的本科教育工作。 第二部分几何学中最重要的概念是“循环”。 在代数几何中，一个循环对应于一个多项式方程组的联立解。在微分几何中，它们以多种方式出现：作为某些微分方程的大规模解，以及作为可微映射的水平集和奇点集。 空间中的曲线和曲面是简单的例子。 具有特定几何形状的周期在现代物理理论中也起着重要作用。在代数学中，圈与其周围空间的基本大尺度几何有关。这一发现揭示了代数圈空间和代数拓扑学基本结构之间的重要关系，并导致了这两个领域的新见解。 这一工作将继续下去。另一个研究领域涉及形成具有特殊几何结构的子集的边界的圈。 它们代表了分析中经典边值问题的非线性版本。这类问题在许多情况下都会出现。 提议者已经制定了相关的重要类，这样的循环问题，在逼近理论和Banach代数。成功的解决将建立一系列新的结果，在复杂的几何，并应导致重要的新见解，在其他几个领域的proposities.A第三部分的建议旨在扩展经典的多能理论非常一般的几何设置。这些几何学包括校准几何学、辛几何学和拉格朗日几何学等等，大量的经典理论已经被证明在这个一般的背景下成立。解决方案的Dirichlet问题相关的蒙日安培型方程将寻求在此设置。从某种严格意义上说，这种研究是对这些几何中出现的特殊循环的研究。 它应该适用于Calabi-Yau流形中的特殊拉格朗日圈，以及G（2）和Spin（7）空间中的结合圈和Cayley圈。 这些后面的主题涉及规范场论和重力在物理学的第四个领域的调查涉及的数学仪器的提议者和R。哈维试图发现周期与它们所居住空间的全球结构之间的微妙关系。 这个装置包含了历史上为此目的而开发的一些最有效的工具，并且它更加通用。这一理论及其应用的进一步发展将继续下去。 这个项目也将关注研究生的发展。学生将成为研究团队的一部分。 还将有一个本科教育的努力，旨在培养数学的独立性和发展互动的环境。