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。第一个是在代数几何形状中自然出现的霍奇柴尔德结构的研究。hochschild同源性和同源物是Bertchalgebra和几何学中必不可少的不变性，PI提议从镇痛的视角来研究它们。第二个是空间的解释示例的研究，该示例具有{\ Emgenus One纤维化}的额外结构。 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猜想。第二个目的是了解平滑品种的Hochschild同源性及其与Poincar \ e配对的关系的抽象配对。这将为Riemann-Roch定理提供新的观点。第三个项目是对小型属性小部分的竞争研究的研究。它在弦理论中具有稳定的奇异性以及对calabi-yau三倍的托雷利问题的理解，它具有应用于研究。特别重要的是对现代理论物理学的应用，特别是在弦理论中。当前的物体将增加我们对在代数几何和物理中重要的空间几何形状的一般理解。