Hessenberg varieties

Flag varieties form an interesting class of homogeneous algebraic varieties. In this talk we consider certain subvarieties of flag varieties, known as Hessenberg varieties. Let $X$ be the complex $n \times n$ matrix of a linear map $X\colon \mathbb{C}^n \rightarrow \mathbb{C}^n$, and let $h\colon\{1, 2, \ldots,n\} \rightarrow \{1, 2, \ldots,n\}$ be a Hessenberg function, i.e. a non-decreasing function satisfying the inequalities $h(j) \geq j$ for $1 \leq j \leq n$. The Hessenberg variety $Hess(X,h)$ associated to $X$ and $h$ is defined as follows:
$$Hess(X,h) = \{V_{\bullet} |XV_i \subseteq V_{h(i)} \text{ for all } 1 \leq i \leq n \}.$$
Particular examples of Hessenberg varieties include full flag varieties and Springer fibers, i.e., the fibers of the Springer resolution of the nilcone in $\mathfrak{gl}(n)$. I will discuss basic properties of these varieties and then explain the relation between their cohomology rings and Schubert polynomials.