Deformation of Bott-Samelson varieties and variations of isotropy structures

TL;DR

This paper investigates how Bott-Samelson varieties associated with certain flag manifolds can deform, revealing automorphisms of Schubert varieties and offering an algebraic proof for flag manifold characterization.

Contribution

It demonstrates that deformations of Bott-Samelson varieties relate to automorphisms of Schubert varieties, leading to a unified algebraic proof of flag manifold characterization.

Findings

Deformations of Bott-Samelson varieties can be explained via automorphisms.

Variations of isotropic structures are linked to these deformations.

Provides an algebraic proof for flag manifold characterization.

Abstract

In the framework of the problem of characterizing complete flag manifolds by their contractions, the complete flags of type $F_4$ and $G_2$ satisfy the property that any possible tower of Bott-Samelson varieties dominating them birationally deforms in a nontrivial moduli. In this paper we illustrate the fact that, at least in some cases, these deformations can be explained in terms of automorphisms of Schubert varieties, providing variations of certain isotropic structures on them. As a corollary, we provide a unified and completely algebraic proof of the characterization of complete flag manifolds in terms of their contractions.