模空间及相关几何不变量的研究

During last three decades, moduli space played very important role in the study of geometry and topology of manifolds, and has become a very popular topic in the study of geometric topology and mathematical physics. In this project, one will study birational symplectic geometry, Gromov-Witten invariant of Hilbert schemes of points on surfaces, orbifold Gromov-Witten invariant, Donaldson-Thomas invariant and quantum K-theory. We expect to prove that Hilbert schemes of points on some surfaces are numerically rationally connected;to obtain the change of Gromov-Witten invariant of Hilbert scheme of points on surfaces under Blow-up of surfaces; to prove the Blow-up correspondence for orbifold absolute/relative Gromov-Witten invariants and apply this correspondence to study the birational classification of orbifolds; to define quantum K-invariants and establish quantum K-theory and its geometric properties for general symplectic manifolds.

模空间在近三十年的几何拓扑研究中发挥了至关重要的作用，是当前几何拓扑及数学物理的研究热点问题之一。本项目计划开展关于双有理辛几何、曲面上点的Hilbert概形的Gromov-Witten不变量、orbifold Gromov-Witten不变量、Donaldson-Thomas不变量及量子K-理论的研究。预期证明若干曲面上点的HIlbert概形的数值双连通性；给出曲面上点的Hilbert概形的Gromov-Witten不变量在曲面Blow-up下的变化；给出orbifold Gromov-Witten不变量在Blow-up下的对应关系并用来研究orbifold的双有理分类；对辛流形定义量子K-不变量并建立起量子K-理论及其性质。

模空间在近三十年的几何拓扑研究中发挥了至关重要的作用，是当前几何拓扑及数学物理的研究热点问题之一。本项目主要研究了曲面上点的Hilbert 概形的Gromov-Witten 不变量、orbifold Gromov-Witten 不变量、Donaldson-Thomas 不变量及6维辛流形的Gromov-Witten 不变量的Blowup公式。 给出了若干曲面上点的HIlbert 概形的 Gromov-Witten不变量的计算； 给出orbifold Gromov-Witten 不变量沿光滑点的Blow-up公式；得到了6维辛流形的高亏格Gromov-Witten不变量的Blowup公式； 证明了Weinstein 猜测对某些乘积辛流形成立；给出了Welschinger不变量的Blowup公式；给出了轨形绝对/相对Gromov-Witten不变量的Blow-up对应。