Topology and geometry of flag bott-samelson varieties and bott towers

Abstract

One of the most natural way to construct a new manifold from a given one is considering fiber bundles. In this paper we take the projective space and the full flag manifold as fibers. Iterating these procedures the former gives Bott-Samelson manifolds, Bott manifolds, and generalized Bott manifolds, and the later produces flag Bott-Samelson manifolds, and flag Bott manifolds. Grossberg and Karshon define a family of complex structures on a Bott-Samelson variety degenerates to a Bott tower, which is a toric variety. This connection defines a virtual polytope, called a twisted cube, which encodes the character of a representation. We study a sufficient and necessary condition for untwistedness of certain twisted cubes. We define flag Bott-Samelson manifolds and flag Bott manifolds, and we show that flag Bott-Samelson manifolds degenerate flag Bott towers, and also this relation gives a combinatorial object which encodes the character of a representation. Moreover we show that Newton-Okounkov bodies of flag Bott-Samelson varieties are generalized string polytopes. On the other hand there is a different extended notion of Bott tower, called a generalized Bott tower introduced by Masuda and Suh. We 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-ups of the former along certain invariant submanifolds. We use the GKM structure of a flag Bott tower together with some toric topological arguments to prove it.