Quantum Cohomology and Birational Geometry

摘要：量子上同调可以被认为是几何对象辛流形的一种代数不变结构。一个二元变换可以被认为是一个辛流形的变换，辛流形是一个在可能发生不连续的小区域之外的同构。这个项目的动机是以下问题：什么是量子上同调的正则变换？也就是说，什么样的辛流形变换能保持量子上同调？首席研究员的早期结果表明，双国变换可能起着规范变换的作用。本项目致力于系统研究这一现象。

Proposal: DMS-0072282Abstract: Quantum cohomology can be thought of as an algebraic invariant structure of geometric objects called symplectic manifolds. A birational transformation can be thought of as a transformation of a symplectic manifold that is an isomorphism outside of a tiny region where discontinuity can happen. The motivation of this project is the following question: What is the canonical transformation of quantum cohomology? Namely, what kind of transformation of symplectic manifolds will preserve quantum cohomology? The principal investigator's early results indicate that birational transformations may play the role of canonical transformation. This project is devoted to the systematic study of this phenomenon.