# Characterizing symplectic Grassmannians by varieties of minimal rational tangents

## TL;DR

This paper characterizes symplectic and odd-symplectic Grassmannians using their varieties of minimal rational tangents (VMRT), establishing their rigidity among Fano manifolds with Picard number 1 and extending Tanaka theory beyond parabolic geometries.

## Contribution

It introduces a generalized Tanaka method applicable to non-parabolic geometries, enabling the characterization and rigidity results for symplectic Grassmannians via VMRT.

## Key findings

- VMRT at a general point determines symplectic Grassmannians
- Symplectic Grassmannians are rigid under Kähler deformation
- Generalized Tanaka theory applies to broader geometric settings

## 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 odd-symplectic Grassmannians, among Fano manifolds of Picard number 1, by their VMRT at a general point and prove their rigidity under global K\"ahler 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 pseudo-concavity type condition that certain vector bundles arising from Spencer complexes have no nonzero sections. The pseudo-concavity type condition is checked by exploiting geometry of minimal rational curves.