2026/07/21 by Cong Ding, Qifeng Li
#math.AG #math.CV
A Schubert variety X0 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 X0 must satisfy Z=g⋅ X0 for some g∈\rm Aut0(X). We say X0 is Schur rigid, if furthermore any subvariety Z on X whose homology class is a multiple r of that of X0 must satisfy Z=g1⋅ X0+⋯+gr⋅ X0 for some g1,⋯ ,gr∈\rm Aut0(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 X0 of subdiagram type, we show that it has Schur rigidity in long root cases unless X0 admits a fiber bundle structure over the projective space.