Rigid Schubert varieties in compact Hermitian symmetric spaces

Abstract

Given a singular Schubert variety�X�w�in a compact Hermitian symmetric space�X, it is a long-standing question to determine when�X�w�is homologous to a smooth variety�Y. We identify those Schubert varieties for which there exist first-order obstructions to the existence of�Y. This extends (independent) work of M. Walters, R. Bryant and J. Hong. Key tools include (i) a new characterization of Schubert varieties that generalizes the well-known description of the smooth Schubert varieties by connected sub-diagrams of a Dynkin diagram and (ii) an algebraic Laplacian (� la Kostant), which is used to analyze the Lie algebra cohomology group associated with the problem.