代数几何和表示论的一些新进展

New advances in algebraic geometry and representation theory

双有理几何在双有理等价意义下对射影代数簇及其推广射影偶进行分类,是代数几何中的一个主要研究方向.代数几何与表示论具有不可分割的紧密联系,辛奇点就是同时涉及到这两个领域的一个重要研究课题.本文在对双有理几何中的基本概念进行简短的介绍后,探讨了典范体积的分布及应用,该不变量在射影偶的有界性问题上发挥关键作用.介绍了辛奇点的形变及量子化,及其在表示论中的应用.

As a major research subfield of algebraic geometry,birational geometry studies the classification of projective algebraic varieties and their generalization as well as projective pairs under birational equivalence.After a brief introduction to the basic concepts in birational geometry,the first part of the survey explores the distribution and applications of the canonical volumes,which constitutes a key invariant in the boundedness problems of projective pairs.On the other hand,algebraic geometry and representation theory secure an inseparable connection,and symplectic singularities qualify as one of the important research topics that involve both fields.The second part of this article focuses on the deformation and quantization of symplectic singularities,as well as their applications in representation theory.