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.

中文翻译：

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用各种最小有理切线表征辛格拉斯曼人

我们证明，如果单规则射影流形的最小有理切线（VMRT）的多样性在一个一般点投影上等价于辛或奇辛格拉斯曼量，则一般最小有理曲线的胚对于胚是双同胚的前辛格拉斯曼算式中的一条一般线。作为一个应用，我们在 Picard 数 1 的 Fano 流形中通过它们的 VMRT 在一般点上表征辛和奇辛 Grassmannians，并证明它们在全局 K\"ahler 变形下的刚性。$G/P$ 的类似结果与十年前，Mok 和 Hong-Hwang 通过使用 Tanaka 理论对抛物线几何获得了长根。当 $G/P$ 与短根相关联时，辛格拉斯曼是最突出的例子 相关的局部微分几何结构不再是抛物线几何，并且由于几个退化特征，无法应用田中理论的标准机制。为了克服这个困难，我们证明了 Tanaka 的方法可以推广到比抛物线几何更广泛的设置，通过假设由 Spencer 复合体产生的某些向量束没有非零截面的伪凹型条件。通过利用最小有理曲线的几何形状来检查伪凹型条件。通过假设由 Spencer 复合体产生的某些向量丛没有非零截面的伪凹型条件。通过利用最小有理曲线的几何形状来检查伪凹型条件。通过假设由 Spencer 复合体产生的某些向量丛没有非零截面的伪凹型条件。通过利用最小有理曲线的几何形状来检查伪凹型条件。