Journal of the Mathematical Society of Japan
Online ISSN : 1881-1167
Print ISSN : 0025-5645
ISSN-L : 0025-5645
Varieties of minimal rational tangents of unbendable rational curves subordinate to contact structures
Jun-Muk Hwang
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2022 Volume 74 Issue 2 Pages 571-590

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Abstract

A nonsingular rational curve 𝐢 in a complex manifold 𝑋 whose normal bundle is isomorphic to π’ͺ_{β„™1}(1)βŠ•π‘ βŠ• π’ͺ_{β„™1}^{βŠ•π‘ž} for some nonnegative integers 𝑝 and π‘ž is called an unbendable rational curve on 𝑋. Associated with it is the variety of minimal rational tangents (VMRT) at a point π‘₯ ∈ 𝐢, which is the germ of submanifolds π’žπΆπ‘₯ βŠ‚ ℙ𝑇π‘₯𝑋 consisting of tangent directions of small deformations of 𝐢 fixing π‘₯. Assuming that there exists a distribution 𝐷 βŠ‚ 𝑇𝑋 such that all small deformations of 𝐢 are tangent to 𝐷, one asks what kind of submanifolds of projective space can be realized as the VMRT π’žπΆπ‘₯ βŠ‚ ℙ𝐷π‘₯. When 𝐷 βŠ‚ 𝑇𝑋 is a contact distribution, a well-known necessary condition is that π’žπΆπ‘₯ should be Legendrian with respect to the induced contact structure on ℙ𝐷π‘₯. We prove that this is also a sufficient condition: we construct a complex manifold 𝑋 with a contact structure 𝐷 βŠ‚ 𝑇𝑋 and an unbendable rational curve 𝐢 βŠ‚ 𝑋 such that all small deformations of 𝐢 are tangent to 𝐷 and the VMRT π’žπΆπ‘₯ βŠ‚ ℙ𝐷π‘₯ at some point π‘₯ ∈ 𝐢 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.

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