Hessenberg varieties是代数几何中的一类重要代数簇。这一类代数簇包含了许多重要的空间，如：旗流形，Peterson variety，permutohedral variety等。Shareshian-Wachs的工作将组合学中著名的Stanley-Stembridge猜想转化为对称群在正则半单Hessenberg variety的上同调环上的表示问题的研究，因此正则半单Hessenberg variety上的代数拓扑研究显得尤为重要。本项目的主要研究内容包括：正则半单Hessenberg variety的代数拓扑，具体为等变上同调环、通常上同调环和Betti数的显式表达；研究正则半单Hessenberg varieties是weak Fano的充分必要条件，重点涉及其上的Chern数的研究。

Hessenberg varieties are a family of important algebraic varieties in algebraic geometry. This family of varieties contains many important spaces, such as the full flag variety, Peterson variety, permutohedral variety and so on. Shareshian-Wachs’ work reduces the famous Stanley-Stembridge conjecture in combinatorics to problems on representations of the symmetric group on the cohomology ring of a regular semisimple Hessenberg variety. Therefore, the study of the algebraic topology of regular semisimple Hessenberg varieties becomes extremely important. In this project we focus on the study of the algebraic topology of a regular semisimple Hessenberg variety, including equivariant cohomology ring, ordinary cohomology ring and Betti numbers. We also study the sufficient and necessary condition for a regular semisimple Hessenberg variety to be weak Fano. This will also involve in studying Chern numbers of a regular semisimple Hessenberg variety.