Abstract
In this paper we consider certain closed subvarieties of the flag variety, known as Hessenberg varieties. These varieties arise in representation theory, algebraic geometry, and combinatorics. We give a connectedness criterion for semisimple Hessenberg varieties that generalizes a criterion given by Anderson and Tymoczko. It also generalizes results of Iveson in type A which prove that all Hessenberg varieties satisfying this criterion are connected. We then show that nilpotent Hessenberg varieties are rationally connected.
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