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

Shintarô Kuroki, Eunjeong Lee, Jongbaek Song and Dong Youp Suh

Vol. 308 (2020), No. 2, 347–392

DOI: 10.2140/pjm.2020.308.347

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.

Keywords

flag Bott tower, flag Bott manifold, generalized Bott manifold, GKM theory, toric manifold, blow-up

Mathematical Subject Classification 2010

Primary: 14M15, 55R10

Secondary: 14M25, 57S25

Milestones

Received: 6 November 2018

Revised: 15 September 2019

Accepted: 6 June 2020

Published: 9 December 2020