Topology of Singular Algebraic Varieties

奇异代数簇的拓扑

• 批准号: 0705050
• 负责人: Anatoly Libgober
• 金额: $ 15.22万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-08-01 至 2012-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0705050&HistoricalAwards=false
• 关键词: Topology; Singular; Algebraic; Varieties

AbstractAward: DMS-0705050Principal Investigator: Anatoly S. LibgoberThe project deals with the study of topological invariants forsingular algebraic varieties with many ideas motivated byphysics. One of its goals is to develop new invariants forsingular spaces extending the theory of elliptic genus ofvarieties with mild singularities and orbifolds which waspresented in joint works with Lev Borisov earlier. This willyields an extension of the previous contributions by many authorson vertex operator algebras associated to manifolds or orbifoldsand application of these invariants to geometric problems. PIplans to develop further generalizations of elliptic genusdetecting the torsion in the cobordisms of special unitarycomplex manifolds as well as use this invariant in classificationand surgery problems of singular spaces. The principalinvestigator will investigate the possibility that the ellipticgenus plays a role in singularity theory beyond classicalapplications in Landau-Ginzburg models associated with weightedhomogeneous polynomials. In particular it is planned to considerthe extensions of elliptic genus of Landau-Ginzburg models inconnection with the Arnold-Steenbrink spectrum of isolatedsingularities. In another direction, PI plans to study thehomotopy groups of non-simply connected quasiprojectivevarieties. In particular the study of characteristic varietiesassociated with homotopy groups controlling among other thingsthe cohomology of local systems will be continued. A.Libgoberwill focus on properties of the Hodge decomposition of the latterand will study the fundamental problem of realization of theseinvariants, i.e. the existence of quasiprojectve varieties withgiven characteristic varieties. As part of this work, PI willinvestigate the restrictions on the dimensions of characteristicvarieties in the case of plane reducible curves generalizingresults in the case of arrangements of lines in complexprojective plane.In general terms the proposal aims to develop new approaches tothe study of geometry of spaces appearing in a variety ofproblems in mathematics and such aspects of theoretical physicsas string theory. A new type of data, called elliptic genus,reflecting geometry and topology of these spaces which appearednaturally in physics will be investigated from a mathematicalperspective and applied to a wide range of problems in geometry,topology and singularity theory. It is planned to study furthergeneralizations of elliptic genus which should have applicationsto fundamental problems of physics. The realization of presenceof infinite dimensional aspects in geometric issues, on which ourwork depends in an essential way, is a novel feature whichemerged in the last 20 years in mathematics and physics. It isplanned to widely disseminate such new viewpoints gainingstrength in contemporary mathematics and physics.

摘要奖：DMS-0705050主要研究者：Anatoly S. Libgober该项目涉及奇异代数簇的拓扑不变量的研究，其中许多想法是由物理学激发的。 它的目标之一是发展新的不变量奇异空间扩展理论的椭圆属的品种温和的奇异性和orbifolds这是提出了联合工程与列夫鲍里索夫较早。 这将产生一个扩展的许多作者以前的贡献顶点算子代数相关联的流形或orbifolds和应用这些不变量的几何问题。PI计划发展椭圆亏格的进一步推广，检测特殊酉复流形的协边中的挠率，以及在奇异空间的分类和外科手术问题中使用这个不变量.主要研究者将研究椭圆亏格在奇异性理论中发挥作用的可能性，超越了与加权齐次多项式相关的Landau-Ginzburg模型中的经典应用。特别是计划将Landau-Ginzburg模型的椭圆亏格的扩展与孤立奇点的Arnold-Steenbrink谱联系起来。在另一个方向上，PI计划研究非单连通拟射影变元的同伦群。特别是与同伦群相关的特征变量的研究，其中包括局部系统的上同调. A. Libgober将着重于后者的Hodge分解的性质，并将研究这些不变量实现的基本问题，即具有给定特征簇的拟投射簇的存在性。 作为这项工作的一部分，PI将调查的限制的尺寸的特征品种的情况下，平面可约曲线概括的结果的情况下，安排的线在复杂的projective plane.In general terms的建议，旨在开发新的方法来研究出现在各种各样的问题，在数学空间的几何和理论物理等方面的弦理论。一种新的数据类型，称为椭圆亏格，反映了这些空间的几何和拓扑，自然出现在物理学中，将从几何的角度进行研究，并应用于广泛的几何，拓扑和奇点理论问题。 计划研究椭圆亏格的进一步推广，这些推广应该在物理学的基本问题中有应用。 实现存在的无限维方面的几何问题，我们的工作依赖于一个基本的方式，是一个新的功能，出现在过去20年的数学和物理学。计划广泛传播这些在当代数学和物理学中日益增强的新观点。