Special lines on contact manifolds
Annales de l'Institut Fourier, Volume 72 (2022) no. 5, pp. 1859-1909.

In a series of two articles Kebekus studied deformation theory of minimal rational curves on contact Fano manifolds. Such curves are called contact lines. Kebekus proved that a contact line through a general point is necessarily smooth and has a fixed standard splitting type of the restricted tangent bundle. In this paper we study singular contact lines and those with special splitting type. We provide restrictions on the families of such lines, and on contact Fano manifolds which have reducible varieties of minimal rational tangents. We also show that the results about singular lines naturally generalise to complex contact manifolds, which are not necessarily Fano, for instance, quasi-projective contact manifolds or compact contact manifolds of Fujiki class 𝒞. In particular, in many cases the dimension of a family of singular lines is at most 2 less than the dimension of the contact manifold.

Dans une série de deux articles, Kebekus a étudié la théorie des déformations des courbes rationnelles minimales sur des variétés de contact de Fano. De telles courbes sot appelées lignes de contact. Kebekus a montré qu’une ligne de contact passant par un oint quelconque est nécessairement régulière et que le type de décomposition du fibré tangent restreint est fixe. Dans le présent article, nous étudions des lignes de contact singulières et celles qui possèdent un type de décomposition particulier. Nous donnons des restrictions sur les familles de telles lignes, et sur les variétés de contact de Fano qui possèdent des variétés réductibles d’espaces tangents rationnels minimaux. Nous montrons aussi que les résultats concernant les lignes singulières s’étendent naturellement aux variétés de contact complexes qui ne sont pas nécessairement de Fano, par exemple les variétés de contact quasi-projectives ou les variétés de contact compactes de la classe de Fujiki 𝒞. En particulier, dans de nombreux cas, la dimension de la famille de lignes singulières est au plus la dimension de la variété de contact moins 2.

Received:
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Accepted:
Published online:
DOI: 10.5802/aif.3510
Classification: 53D10,  14J45,  14M20,  32Q57,  14M22
Keywords: complex contact manifold, minimal rational curves, contact lines, VMRT, manifolds of Fujiki class 𝒞
Buczyński, Jarosław 1, 2; Kapustka, Grzegorz 3; Kapustka, Michał 1

1 Institute of Mathematics of the Polish Academy of Sciences, ul. Śniadeckich 8, 00-656 Warsaw, Poland
2 and Faculty of Mathematics, Computer Science and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland
3 Faculty of Mathematics and Computer Science of the Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Buczyński, Jarosław; Kapustka, Grzegorz; Kapustka, Michał. Special lines on contact manifolds. Annales de l'Institut Fourier, Volume 72 (2022) no. 5, pp. 1859-1909. doi : 10.5802/aif.3510. https://aif.centre-mersenne.org/articles/10.5802/aif.3510/

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