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October 2021 Characterizing symplectic Grassmannians by varieties of minimal rational tangents
Jun-Muk Hwang, Qifeng Li
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J. Differential Geom. 119(2): 309-381 (October 2021). DOI: 10.4310/jdg/1632506422

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.

Funding Statement

The authors are supported by National Researcher Program 2010-0020413 of NRF.

Citation

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Jun-Muk Hwang. Qifeng Li. "Characterizing symplectic Grassmannians by varieties of minimal rational tangents." J. Differential Geom. 119 (2) 309 - 381, October 2021. https://doi.org/10.4310/jdg/1632506422

Information

Received: 10 December 2018; Accepted: 7 July 2020; Published: October 2021
First available in Project Euclid: 27 September 2021

Digital Object Identifier: 10.4310/jdg/1632506422

Subjects:
Primary: 14M17, 14M22, 32G05, 53B15, 53C15

Rights: Copyright © 2021 Lehigh University

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Vol.119 • No. 2 • October 2021
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