Characterizing symplectic Grassmannians by varieties of minimal rational tangents

Abstract

We show that if the variety of minimal rational tangents (VMRT) of a uniruled projective manifold at a general point is projectively equivalent to that of a symplectic or an odd-symplectic Grassmannian, the germ of a general minimal rational curve is biholomorphic to the germ of a general line in a presymplectic Grassmannian. As an application, we characterize symplectic and oddsymplectic Grassmannians, among Fano manifolds of Picard number $1$, by their VMRT at a general point and prove their rigidity under global Kähler deformation. Analogous results for $G/P$ associated with a long root were obtained by Mok and Hong–Hwang a decade ago by using Tanaka theory for parabolic geometries. When $G/P$ is associated with a short root, for which symplectic Grassmannians are most prominent examples, the associated local differential geometric structure is no longer a parabolic geometry and standard machinery of Tanaka theory cannot be applied because of several degenerate features. To overcome the difficulty, we show that Tanaka’s method can be generalized to a setting much broader than parabolic geometries, by assuming a vanishing condition that certain vector bundles arising from Spencer complexes have no nonzero sections. In our application of this method to the characterization of symplectic (or odd-symplectic) Grassmannians, this vanishing condition is checked by exploiting geometry of minimal rational curves.