Gromov-Witten 理论及相关问题研究

Gromov-Witten invariant is one of the most important invariants in geometry and topology, and played very important role in the study of goemetry and topology of manifolds. Now Gromov-Witten theory has become very popular in geometry and Physics. In this project, one will study the symplectic rationally connectedness of symplectic manifolds, Donaldson-Thomas theory and the naturality of quantum cohomology ring. We expect to prove the symplectic rationally connectedness of Del Pezzo varieties and the Hilbert schemes of points of the projective plane; to find the relation between the Donaldson-Thomas theory of the P1-bundle over a surface and the nested Hilbert scheme of points of the surface; to describe the change of quantum cohomology ring under Blowup.

Gromov-Witten不变量是目前最重要的几何拓扑不变量之一，它在几何拓扑的研究中发挥了至关重要的作用，目前已发展成为几何与物理的研究热点问题。本项目计划开展关于辛流形的辛有理连通性、Donaldson-Thomas理论和量子上同调环的自然性等研究。预期证明Del Pezzo 簇，Fano簇及射影平面的点的Hilbert概型的辛有理连通性；给出曲面上P1-丛的Donaldson-Thomas理论与曲面的嵌套Hilbert概型的Gromov-Witten不变量之间的关系；给出量子上同调环在Blowup下的变化。

Gromov-Witten不变量是辛拓扑与数学物理最重要的几何拓扑不变量之一。本项目主要研究Gromov-Witten不变量性质及如何应用Gromov-Witten不变量来刻画辛流形的几何拓扑。本项目获得的重要结果包括：Welschinger不变量的Blowup公式，高亏格Gromov-Witten 不变量的Blowup公式，有理曲面的2-点Hilbert概型的辛有理连通性， 轨形相对/绝对Gromov-Witten不变量在加权Blowup下的对应及辛轨形的uniruled性质以及del Pezzo曲面的Gamma猜测I等。这些结果对进一步研究辛流形的几何拓扑及l量子上同调的自然性的研究具有重要意义。