实枚举不变量的研究

Gromov-Witten不变量是辛几何与数学物理领域最重要的几何拓扑不变量之一。Welschinger不变量、实Gromov-Witten不变量作为其实的版本受到广泛关注，热带加细不变量随之也被发现，这些实枚举不变量逐步发展成为代数几何、辛几何与数学物理的重要研究方向。本项目将研究实Gromov-Witten不变量对实结构的依赖性，热带加细不变量的退化性质，以及定义4维实辛流形上高亏格Welschinger不变量并得到相应递归公式。在借鉴已有的Gromov-Witten不变量退化公式，热带几何和横截性理论等方法的基础上，还需要利用Dehn twist刻画实结构的变化，分析热带曲线的退化类型，以及构造可以排除曲线坏的退化类型的几何条件以探索解决项目中的问题。本项目的研究对理解实辛流形的几何拓扑具有重要意义。

Gromov-Witten invariant is one of the most important geometric and topological invariants in the field of symplectic geometry and mathematical physics. Welschinger invariant and real Gromov-Witten invariant were the real counterparts of Gromov-Witten invariants and aroused the interests of many mathematicians. Tropical refined invariant was also discovered later. The research on these real enumerative invariants becomes an important area in algebraic geometry, symplectic geometry and mathematical physics. In this project, we will study real Gromov-Witten invariant's dependence on real structures, degeneration properties of tropical refined invariants and the definition of higher genus Welschinger invariants on real symplectic 4-manifolds. Except for degeneration formula of Gromov-Witten invariants, tropical geometry and transversality theory, we also need to describe the change of real structures via Dehn twist and analyse the degeneration type of tropical curves. In order to solve the problems of this project, we also have to construct a proper geometric condition to exclude the bad degeneration of real curves. The study in this project is of great significance in understanding the geometry and topology of real symplectic manifolds.