镜对称及双有理代数几何背景下的高维 Calabi-Yau 簇

项目名称： 镜对称及双有理代数几何背景下的高维 Calabi-Yau 簇

项目编号： No.11601015

项目类型： 青年科学基金项目

立项/批准年度： 2017

项目学科： 数理科学和化学

项目作者： 李展

作者单位： 北京大学

项目金额： 15万元

中文摘要： 三维 Calabi-Yau 簇是镜对称理论的重要研究对象. 此外,高维 Calabi-Yau 簇是高维代数簇分类中相对缺失的部分... 本项目将研究 Calabi-Yau 簇在镜对称中的导出范畴等价关系，以及 Hochschild 同调与弦论上同调的关系. 同时，我们将研究有纤维结构的三维 Calabi-Yau 簇的有界性，以及 Calabi-Yau 簇在小形变意义下的双有理性质等... 通过对这一系列问题的深入研究，有望促进镜对称理论与双有理代数几何两方面的发展，并使它们之间交叉作用，互相影响.

中文关键词： Calabi-Yau；簇；镜对称；双有理代数几何；双有理有界性；导出范畴

英文摘要： Three dimensional Calabi-Yau varieties are fundamental objects for mirror symmetry. Besides, higher-dimensional Calabi-Yau varieties are the relatively missing part in the classifications of higher-dimensional algebraic varieties...This proposal aims to research on derived equivalence of Calabi-Yau varieties in mirror symmetry, and the relations between Hochschild homology and string cohomology. Meanwhile, we will investigate on the boundedness problem of fibered Calabi-Yau threefolds as well as birational properties of Calabi-Yau varieties under small deformations...These investigations will promote mirror symmetry and birational geometry simultaneously, and let them interact with each other.

英文关键词： Calabi-Yau varieties;mirror symmetry;birational geometry;birational boundedness;derived categories