Jaehyun Hong, Seoul National University, Korea

Classification of smooth Schubert varieties in a rational homogeneous manifold of Picard number one

Abstract

A Schubert variety of a rational homogeneous manifold G/P is the closure of an orbit of a Borel subgroup of G. By the Bruhat decomposition the homology classes of Schubert varieties generate the homology space of G/P. Schubert varieties are generally singular and an explicit determination of the singular locus of a Schubert variety is still an open question.

A normal variety with an action of a reductive group is said to be horospherical if it has an open dense orbit which is a torus bundle over a rational homogenous manifold. Up to now all known examples of smooth Schubert varieties are horospherical. In this talk, we show that a smooth Schubert variety of a rational homogeneous manifold of Picard number one is horospherical and determine all smooth Schubert varieties of rational homogeneous manifolds of Picard number one.