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

TL;DR

This paper explores the relationship between flag Bott towers and generalized Bott towers, showing that flag Bott towers are GKM manifolds and their generic orbit closures relate to blow-ups of generalized Bott towers.

Contribution

It demonstrates that flag Bott towers are GKM manifolds and establishes a connection between generic orbit closures and blow-ups of generalized Bott towers.

Findings

Flag Bott towers are GKM manifolds.

Closure of generic orbits in flag Bott towers are blow-ups of generalized Bott towers.

Uses GKM theory and toric geometry to analyze these structures.

Abstract

To a direct sum of holomorphic line bundles, we can associate two fibrations, whose fibers are, respectively, the corresponding full flag manifold and the corresponding projective space. Iterating these procedures gives, respectively, a flag Bott tower and a generalized Bott tower. It is known that a generalized Bott tower is a toric manifold. However a flag Bott tower is not toric in general but we show that it is a GKM manifold, and we also show that for a given generalized Bott tower we can find the associated flag Bott tower so that the closure of a generic torus orbit in the latter is a blow-up of the former along certain invariant submanifolds. We use GKM theory together with toric geometric arguments.