Hochschild theory in algebraic geometry

这个项目有两个主题。首先是对代数几何中自然出现的hochschild结构的研究。Hochschild同调和上同调是代数和几何中的重要不变量，本文提出从代数几何的角度研究它们。第二个是研究具有{\emgenus one fibration}额外结构的空间的明确例子。这些空间在古典和现代代数几何中无处不在，PI期望通过使用派生范畴的技术可以获得对其几何的新见解。本提案包含三个项目。第一个项目是关于寻找Hochschild上同环轨道上的乘法的显式公式，并应用于阮氏猜想。第二部分旨在理解光滑变体Hochschild同调上的抽象Mukai配对及其与微分形式上的庞加莱配对的关系。这将为黎曼-洛克定理提供一个新的视角。第三个项目是具有多节小度的一属纤维的研究。它可以应用于弦理论中稳定奇点的研究，以及对Calabi-Yau三倍的Torelli问题的理解。代数几何是对多项式方程解的几何研究，在过去几年中取得了重大进展，特别是在其他科学领域的应用方面。特别重要的是在现代理论物理学中的应用，特别是在弦理论中的应用。本项目将增加我们对代数几何和物理中一些重要空间的几何的一般理解。

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.