CLASSIFICATION OF SMOOTH SCHUBERT VARIETIES IN THE SYMPLECTIC GRASSMANNIANS

Abstract

Abstract. A Schubert variety in a rational homogeneous variety G/P isdefined by the closure of an orbit of a Borel subgroup Bof G. In general,Schubert varieties are singular, and it is an old problem to determinewhich Schubert varieties are smooth. In this paper, we classify all smoothSchubert varieties in the symplectic Grassmannians. 1. IntroductionA rational homogeneous manifold S= G/Pis a projective manifold, wherea connected complex semisimple group Gacts transitively. Under the actionof a Borel subgroup Bof G, Shas finitely many orbits. The closure of a B-orbit in Sis called a Schubert variety of S. In general, Schubert varieties aresingular, and it is an old problem to determine which Schubert varieties aresmooth. Lakshmibai-Weyman and Brion-Polo have studied the singular lociof Schubert varieties of S, when Sis a compact Hermitian symmetric space([9] and [2]). In particular, they showed that in this case any smooth Schubertvariety in Sis a homogeneous submanifold of Sassociated to a subdiagramof the marked Dynkin diagram of S. For example, a Schubert variety of theGrassmannian Gr(k,V) of k-subspaces in a vector space V is smooth if andonly if it is a linearly embedded sub-Grassmannian.More generally, when Sis associated to a long simple root, we have:Theorem 1.1 (Proposition 3.7 of Hong-Mok [4]). Let S= G/P be a rationalhomogeneous manifold associated to a long simple root. Then any smooth Schu-bert variety in Sis a homogeneous submanifold of Sassociated to a subdiagramof the marked Dynkin diagram of S.On the other hand, when Sis associated to a short simple root, there is asmooth Schubert variety that is not homogeneous. Let V be a vector spacewith a symplectic form ω, i.e., a nondegenerate skew-symmetric bilinear form.