Characterization of smooth Schubert varieties in rational homogeneous manifolds of Picard number 1

摘要

In a series of works one of the authors has developed with J.-M. Hwang a geometric theory of uniruled projective manifolds basing on the study of varieties of minimal rational tangents, and the geometric theory has especially been applied to rational homogeneous manifolds of Picard number 1. In Mok [Astérisque 322, pp. 151-205] and Hong-Mok [J. Diff. Geom. 86(2010), pp. 539-567] the authors have started the study of uniruled projective subvarieties, and a method was developed for characterizing certain subvarieties of rational homogeneous manifolds. The method relies on non-equidimensional Cartan-Fubini extension and a notion of parallel transport of varieties of minimal rational tangents. In the current article we apply the notion of parallel transport to a characterization of smooth Schubert varieties of rational homogeneous manifolds of Picard number 1. Given a pair (S, SO) consisting of a rational homogeneous manifold S of Picard number 1 and a smooth Schubert variety S O of S, where no restrictions are placed on SO when S = G/P is associated to a long root (while necessarily some cases have to be excluded when S is associated to a short root), we prove that any subvariety of S having the same homology class as SO must be gSO for some g ∈ Aut(S). We reduce the problem first of all to a characterization of local deformations St of SO as a subvariety of S. By Kodaira stability, St is uniruled by minimal rational curves of S lying on St. We establish a biholomorphism between St and S O which extends to a global automorphism by reconstructing S t by means of a repeated use of parallel transport of varieties of minimal rational tangents along minimal rational curves issuing from a general base point. Our method is applicable also to the case of singular Schubert varieties provided that there exists a minimal rational curve on the smooth locus of the variety.