2019/01/02 by Jun-Muk Hwang, Qifeng Li, Hwang, Jun-Muk +1 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.1901.00357
openalex publication_date 2019/01/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that if the variety of minimal rational tangents (VMRT) of a uniruled\nprojective manifold at a general point is projectively equivalent to that of a\nsymplectic or an odd-symplectic Grassmannian, the germ of a general minimal\nrational curve is biholomorphic to the germ of a general line in a\npresymplectic Grassmannian. As an application, we characterize symplectic and\nodd-symplectic Grassmannians, among Fano manifolds of Picard number 1, by their\nVMRT at a general point and prove their rigidity under global K "ahler\ndeformation. Analogous results for G/P associated with a long root were\nobtained by Mok and Hong-Hwang a decade ago by using Tanaka theory for\nparabolic geometries. When G/P is associated with a short root, for which\nsymplectic Grassmannians are most prominent examples, the associated local\ndifferential geometric structure is no longer a parabolic geometry and standard\nmachinery of Tanaka theory cannot be applied because of several degenerate\nfeatures. To overcome the difficulty, we show that Tanaka's method can be\ngeneralized to a setting much broader than parabolic geometries, by assuming a\npseudo-concavity type condition that certain vector bundles arising from\nSpencer complexes have no nonzero sections. The pseudo-concavity type condition\nis checked by exploiting geometry of minimal rational curves.\n