Flag Bott manifolds and the toric closure of a generic orbit associated to a generalized Bott manifold (Q827445)

The authors define flag Bott towers \(\{F_j\,|\,0\leq j\leq m\}\) as sequences of the full flag fibrations \(Fl(n_{j}+1)\hookrightarrow F_j\to F_{j-1}\), where \(F_j\) is the flagification of a sum of \(n_{j}+1\) many complex line bundles over \(F_{j-1}\). Each \(F_j\) is called a flag Bott manifold. One of the main aims of the article is to study torus actions on flag Bott manifolds. Let \(F_m\) be an \(m\)-stage flag Bott manifold. Observe that there is an effective action of the complex torus \(H\) of dimension \(\sum\limits_{j=1}^{m}n_j\) on \(F_m\). Consider \(F_m\) with the restricted action of the compact torus \(T\) of dimension \(\sum\limits_{j=1}^{m}n_{j}\). A flag Bott manifold is not a toric manifold in general; however, the authors proved that \((F_m,T)\) is a GKM manifold. They also consider the class of generalized Bott towers \(\{B_j\,|\,0\leq j\leq m\}\), where \(B_j\) is the projectivization of the sum of \(n_{j}+1\) many complex line bundles instead of two line bundles, and define the associated flag Bott tower to each generalized Bott tower. Let \(B_m\) be an \(m\)-stage generalized Bott manifold, and \(F_m\) its associated flag Bott manifold. Then the authors showed that the closure of a generic orbit of \(H\)-action in \(F_m\) is the blow-up of \(B_m\) along certain invariant submanifolds.