Rigid Schubert varieties in compact Hermitian symmetric spaces

TL;DR

This paper investigates when singular Schubert varieties in compact Hermitian symmetric spaces can be homologous to smooth varieties, identifying obstructions through new characterizations and algebraic tools.

Contribution

It introduces a new characterization of Schubert varieties and uses an algebraic Laplacian to analyze obstructions to smooth homologous varieties.

Findings

Identifies Schubert varieties with first-order obstructions to smooth homologs.

Extends previous work by Walters, Bryant, and Hong.

Provides algebraic criteria for smoothability of singular Schubert varieties.

Abstract

Given a singular Schubert variety Z in a compact Hermitian symmetric space it is a longstanding question to determine when Z 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 (a la Kostant), which is used to analyze the Lie algebra cohomology group associated to the problem.