Hochschild theory in algebraic geometry

代数几何中的霍克希尔德理论

• 批准号: 0556042
• 负责人: Andrei Caldararu
• 金额: $ 11.83万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-01 至 2009-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0556042&HistoricalAwards=false
• 关键词: Hochschild; theory; algebraic; geometry

There are two themes to this project. The first is the study ofHochschild structures that arise naturally in algebraic geometry.Hochschild homology and cohomology are essential invariants in bothalgebra and geometry, and the PI proposes to study them from analgebraic-geometric point of view. The second is the study ofexplicit example of spaces which have the extra structure of {\emgenus one fibration}. These spaces are ubiquitous in classical andmodern algebraic geometry, and the PI expects that by using thetechniques of derived categories new insights can be gained into theirgeometry.There are three projects contained in the present proposal.The first project is concerned with finding explicit formulas for themultiplication in the Hochschild cohomology ring of orbifolds, withapplications to the Ruan conjecture. The second aims at understandingthe abstract Mukai pairing on the Hochschild homology of smooth varieties,and its relationship to the Poincar\'e pairing on differential forms.This would provide a new perspective on the Riemann-Roch theorem.The third project is a study of genus one fibrations withmulti-sections of small degree. It has applications to the study ofstable singularities in string theory, as well as to the understandingof the Torelli problem for Calabi-Yau threefolds.Algebraic geometry, which is the geometric study of solutions ofpolynomial equations, has seen in the last few years majordevelopments, especially in terms of its applications in other fieldsof science. Of particular importance are applications to moderntheoretical physics, in particular in string theory. The presentproject will increase our general understanding of the geometry ofsome of the spaces that are important in algebraic geometry andphysics.

这个项目有两个主题。首先是研究代数几何中自然出现的Hochschild结构，Hochschild同调和上同调是代数和几何中必不可少的不变量，PI建议从代数几何的角度来研究它们。第二部分是研究具有{\emgenus one fibration}额外结构的空间的显式例子。这些空间在经典和现代代数几何中是普遍存在的，PI期望通过使用导出范畴的技术可以获得对它们的几何的新的见解。本建议包含三个项目：第一个项目是关于在orbifolds的Hochschild上同调环中寻找乘法的显式公式，并应用于Ruan猜想。第二个项目旨在理解光滑簇的Hochschild同调上的抽象Mukai配对，以及它与微分形式上的Poincar 'e配对的关系，这将为Riemann-Roch定理提供一个新的视角;第三个项目是研究具有多个小次截面的亏格1纤维化。它可以应用于弦理论中稳定奇点的研究，也可以应用于理解卡-丘三重的Torelli问题。代数几何是对多项式方程解的几何研究，在过去的几年里有了重大的发展，特别是在其他科学领域的应用方面。特别重要的是应用到现代理论物理学，特别是在弦理论。目前的项目将增加我们的一般理解几何的一些空间是重要的代数几何和物理。