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Hwang, Jun Muk
복소기하학 연구단
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Characterizing symplectic Grassmannians by varieties of minimal rational tangents

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Title
Characterizing symplectic Grassmannians by varieties of minimal rational tangents
Author(s)
Jun-Muk Hwang; Qifeng Li
Publication Date
2021-10
Journal
Journal of Differential Geometry, v.119, no.2, pp.309 - 381
Publisher
International Press, Inc.
Abstract
© 2021 International Press of Boston, Inc.. All rights reserved.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 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.
URI
https://pr.ibs.re.kr/handle/8788114/10912
DOI
10.4310/jdg/1632506422
ISSN
0022-040X
Appears in Collections:
Center for Complex Geometry (복소기하학 연구단) > 1. Journal Papers (저널논문)
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