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Hwang, Jun Muk
복소기하학 연구단
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Varieties of minimal rational tangents of unbendable rational curves subordinate to contact structures

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Title
Varieties of minimal rational tangents of unbendable rational curves subordinate to contact structures
Author(s)
Jun-Muk Hwang
Publication Date
2022-04
Journal
Journal of the Mathematical Society of Japan, v.74, no.2, pp.571 - 590
Publisher
Mathematical Society of Japan
Abstract
© 2022 The Mathematical Society of JapanA nonsingular rational curve C in a complex manifold X whose normal bundle is isomorphic to OP1 (1)p OP 1q for some nonnegative integers p and q is called an unbendable rational curve on X. Associated with it is the variety of minimal rational tangents (VMRT) at a point x ∈ C, which is the germ of submanifolds CxC ⊂ PTxX consisting of tangent directions of small deformations of C fixing x. Assuming that there exists a distribution D ⊂ TX such that all small deformations of C are tangent to D, one asks what kind of submanifolds of projective space can be realized as the VMRT CxC ⊂ PDx. When D ⊂ TX is a contact distribution, a well-known necessary condition is that CxC should be Legendrian with respect to the induced contact structure on PDx. We prove that this is also a sufficient condition: we construct a complex manifold X with a contact structure D ⊂ TX and an unbendable rational curve C ⊂ X such that all small deformations of C are tangent to D and the VMRT CxC ⊂ PDx at some point x ∈ C is projectively isomorphic to an arbitrarily given Legendrian submanifold. Our construction uses the geometry of contact lines on the Heisenberg group and a technical ingredient is the symplectic geometry of distributions the study of which has originated from geometric control theory.
URI
https://pr.ibs.re.kr/handle/8788114/12303
DOI
10.2969/JMSJ/85868586
ISSN
0025-5645
Appears in Collections:
Center for Complex Geometry (복소기하학 연구단) > 1. Journal Papers (저널논문)
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