  For a connected, simply-connected complex simple algebraic group $G$, we
examine a class of Hessenberg varieties associated with the minimal nilpotent
orbit. In particular, we compute the Poincar\'{e} polynomials and irreducible
components of these varieties in Lie type $A$. Furthermore, we show these
Hessenberg varieties to be GKM with respect to the action of a maximal torus
$T\subseteq G$. The corresponding GKM graphs are then explicitly determined.
Finally, we present the ordinary and $T$-equivariant cohomology rings of our
varieties as quotients of those of the flag variety.

Algebraic Geometry;Representation Theory