Homological rigidity and Schur rigidity of Schubert varieties in rational homogeneous spaces

Abstract

A Schubert variety $X_0$ on a rational homogenous space $X=G/P$ is said to be homologically rigid, if any subvariety $Z$ on $X$ representing the same homology class with $X_0$ must satisfy $Z=g\cdot X_0$ for some $g\in \mathrm{Au}{\mathrm{t}}_0(X)$. We say $X_0$ is Schur rigid, if furthermore any subvariety $Z$ on $X$ whose homology class is a multiple $r$ of that of $X_0$ must satisfy $Z=g_1\cdot X_0+\cdots +g_r\cdot X_0$ for some $g_1,\cdots ,g_r\in \mathrm{Au}{\mathrm{t}}_0(X)$. Homological rigidity and Schur rigidity of Schubert varieties in rational homogeneous spaces of Picard number one have been well studied in extensive literature. In this paper, we study both rigidity problems of Schubert varieties in rational homogeneous spaces of higher Picard numbers. We show that in the long root cases, including all cases when $G$ is of type $ADE$, smooth Schubert varieties have homological rigidity. Besides, we give the complete list of Schubert varieties of subdiagram type with/without homological rigidity. Furthermore, for a Schubert variety $X_0$ of subdiagram type, we show that it has Schur rigidity in long root cases unless $X_0$ admits a fiber bundle structure over the projective space.