no. 5
Special lines on contact manifolds
[Droites spéciales sur variétés de contact]
Buczyński, Jarosław1, 2 ; Kapustka, Grzegorz3 ; Kapustka, Michał1
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
Annales de l'Institut Fourier, Tome 72 (2022) no. 5, pp. 1859-1909.

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.

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.

Reçu le :
Révisé le :
Accepté le :
Publié le :
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 $\mathcal{C}$
Mot clés : variétés complexes, variétés de contact, courbe rationnelles minimales, droites de contact, variété des tangentes aux courbes rationnelles minimales, variétés de la classe $\mathcal{C}$ de Fujiki

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
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Buczyński, Jarosław; Kapustka, Grzegorz; Kapustka, Michał. Special lines on contact manifolds. Annales de l'Institut Fourier, Tome 72 (2022) no. 5, pp. 1859-1909. doi : 10.5802/aif.3510. https://aif.centre-mersenne.org/articles/10.5802/aif.3510/

[1] Araujo, Carolina; Druel, Stéphane On Fano foliations, Adv. Math., Volume 238 (2013), pp. 70-118 | DOI | MR | Zbl

[2] Araujo, Carolina; Druel, Stéphane On codimension 1 del Pezzo foliations on varieties with mild singularities, Math. Ann., Volume 360 (2014) no. 3-4, pp. 769-798 | DOI | MR | Zbl

[3] Barlet, Daniel Espace analytique réduit des cycles analytiques complexes compacts d’un espace analytique complexe de dimension finie, Fonctions de plusieurs variables complexes, II (Sém. François Norguet, 1974–1975) (Lecture Notes in Mathematics), Volume 482, Springer, 1975, pp. 1-158 | MR | Zbl

[4] Barlet, Daniel Note on rational 1-dimensional compact cycles (2016) | arXiv

[5] Barlet, Daniel; Magnússon, Jón Cycles analytiques complexes. I. Théorèmes de préparation des cycles, Contributions in Mathematical and Computational Sciences, 22, Société Mathématique de France, 2014, 525 pages | MR

[6] Beauville, Arnaud Fano contact manifolds and nilpotent orbits, Comment. Math. Helv., Volume 73 (1998) no. 4, pp. 566-583 | DOI | MR | Zbl

[7] Beltrametti, Mauro C.; Sommese, Andrew J. The adjunction theory of complex projective varieties, de Gruyter Expositions in Mathematics, 16, Walter de Gruyter, 1995, xxii+398 pages | DOI | MR

[8] Bogomolov, Fedor; McQuillan, Michael Rational curves on foliated varieties, Foliation theory in algebraic geometry (Simons Symposia), Springer, 2016, pp. 21-51 | DOI | MR | Zbl

[9] Brunella, Marco Uniformisation of foliations by curves, Holomorphic dynamical systems (Lecture Notes in Mathematics), Volume 1998, Springer, 2010, pp. 105-163 | DOI | MR | Zbl

[10] Buczyński, Jarosław Algebraic Legendrian varieties, Diss. Math., Volume 467 (2009), p. 86 (PhD thesis, Institute of Mathematics, Warsaw University, 2008.) | DOI | MR | Zbl

[11] Buczyński, Jarosław Duality and integrability on contact Fano manifolds, Doc. Math., Volume 15 (2010), pp. 821-841 | MR | Zbl

[12] Buczyński, Jarosław; Moreno, Giovanni Complex contact manifolds, varieties of minimal rational tangents, and exterior differential systems, Geometry of Lagrangian Grassmannians and nonlinear PDEs (Banach Center Publications), Volume 117, Polish Academy of Sciences, 2019, pp. 145-176 | MR | Zbl

[13] Buczyński, Jarosław; Peternell, Thomas Contact Moishezon threefolds with second Betti number one, Arch. Math., Volume 98 (2012) no. 5, pp. 427-431 | DOI | MR | Zbl

[14] Buczyński, Jarosław; Wiśniewski, Jarosław; Weber, Andrzej Algebraic torus actions on contact manifolds (2018) (to appear in J. Differ. Geom.) | arXiv

[15] Campana, Frédéric; Păun, Mihai Foliations with positive slopes and birational stability of orbifold cotangent bundles, Publ. Math., Inst. Hautes Étud. Sci., Volume 129 (2019), pp. 1-49 | DOI | MR | Zbl

[16] Demailly, Jean-Pierre On the Frobenius integrability of certain holomorphic p-forms, Complex geometry (Göttingen, 2000), Springer, 2002, pp. 93-98 | DOI | MR | Zbl

[17] Déserti, Julie; Cerveau, Dominique Feuilletages et actions de groupes sur les espaces projectifs, Mém. Soc. Math. Fr., Nouv. Sér. (2005) no. 103, p. vi+124 | MR | Zbl

[18] Dieudonné, Jean Une généralisation des espaces compacts, J. Math. Pures Appl., Volume 23 (1944), pp. 65-76 | MR | Zbl

[19] Douady, Adrien Le problème des modules pour les sous-espaces analytiques compacts d’un espace analytique donné, Ann. Inst. Fourier, Volume 16 (1966) no. 1, pp. 1-95 | DOI | MR | Zbl

[20] Druel, Stéphane Structures de contact sur les variétés algébriques de dimension 5, C. R. Math. Acad. Sci. Paris, Volume 327 (1998) no. 4, pp. 365-368 | DOI | MR | Zbl

[21] Fischer, Gerd Complex analytic geometry, Lecture Notes in Mathematics, 538, Springer, 1976, vii+201 pages | DOI | MR

[22] Frantzen, Kristina; Peternell, Thomas On the bimeromorphic geometry of compact complex contact threefolds, Classification of algebraic varieties (EMS Series of Congress Reports), European Mathematical Society, 2011, pp. 277-288 | DOI | MR | Zbl

[23] Fujiki, Akira Closedness of the Douady spaces of compact Kähler spaces, Publ. Res. Inst. Math. Sci., Volume 14 (1978/79) no. 1, pp. 1-52 | DOI | MR | Zbl

[24] Görtz, Ulrich; Wedhorn, Torsten Algebraic geometry I. Schemes with examples and exercises, Advanced Lectures in Mathematics (ALM), Vieweg+Teubner, 2010, viii+615 pages | DOI | MR

[25] Several complex variables. VII. Sheaf-theoretical methods in complex analysis (Grauert, H.; Peternell, Th.; Remmert, R., eds.), Encyclopaedia of Mathematical Sciences, 74, Springer, 1994, vi+369 pages | DOI | MR | Zbl

[26] Grothendieck, Alexander Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III, Publ. Math., Inst. Hautes Étud. Sci. (1966) no. 28, p. 255 | MR | Zbl

[27] Hartshorne, Robin Algebraic geometry, Graduate Texts in Mathematics, 52, Springer, 1977, xvi+496 pages | DOI | MR

[28] Hironaka, Heisuke Flattening theorem in complex-analytic geometry, Am. J. Math., Volume 97 (1975), pp. 503-547 | DOI | MR | Zbl

[29] Hwang, Jun-Muk; Manivel, Laurent Quasi-complete homogeneous contact manifold associated to a cubic form, Vector bundles and complex geometry (Contemporary Mathematics), Volume 522, American Mathematical Society, 2010, pp. 105-112 | DOI | MR | Zbl

[30] Hwang, Jun-Muk; Mok, Ngaiming Birationality of the tangent map for minimal rational curves, Asian J. Math., Volume 8 (2004) no. 1, pp. 51-63 | DOI | MR | Zbl

[31] Ilyashenko, Yulij; Yakovenko, Sergei Lectures on analytic differential equations, Graduate Studies in Mathematics, 86, American Mathematical Society, 2008, xiv+625 pages | MR

[32] Kebekus, Stefan Lines on contact manifolds, J. Reine Angew. Math., Volume 539 (2001), pp. 167-177 | MR | Zbl

[33] Kebekus, Stefan Families of singular rational curves, J. Algebr. Geom., Volume 11 (2002) no. 2, pp. 245-256 | DOI | MR | Zbl

[34] Kebekus, Stefan Lines on complex contact manifolds. II, Compos. Math., Volume 141 (2005) no. 1, pp. 227-252 | DOI | MR | Zbl

[35] Kebekus, Stefan; Kovács, Sándor J. Are rational curves determined by tangent vectors?, Ann. Inst. Fourier, Volume 54 (2004) no. 1, pp. 53-79 | DOI | MR | Zbl

[36] Kebekus, Stefan; Peternell, Thomas; Sommese, Andrew J.; Wiśniewski, Jarosław Projective contact manifolds, Invent. Math., Volume 142 (2000) no. 1, pp. 1-15 | DOI | MR | Zbl

[37] Kollár, János Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 32, Springer, 1996, viii+320 pages | DOI | MR

[38] Lazarsfeld, Robert Positivity in algebraic geometry. I. Classical setting: line bundles and linear series, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 48, Springer, 2004, xviii+387 pages | MR

[39] LeBrun, Claude; Salamon, Simon M. Strong rigidity of positive quaternion-Kähler manifolds, Invent. Math., Volume 118 (1994) no. 1, pp. 109-132 | DOI | MR | Zbl

[40] Lee, Wanseok; Park, Euisung; Schenzel, Peter On the classification of non-normal cubic hypersurfaces, J. Pure Appl. Algebra, Volume 215 (2011) no. 8, pp. 2034-2042 | DOI | MR | Zbl

[41] Łojasiewicz, Stanisław Introduction to complex analytic geometry, Birkhäuser, 1991, xiv+523 pages (translated from the Polish by Maciej Klimek) | DOI | MR

[42] Magnússon, Jón A global morphism from the Douady space to the cycle space, Math. Scand., Volume 101 (2007) no. 1, pp. 19-28 | DOI | MR | Zbl

[43] Mella, Massimiliano Existence of good divisors on Mukai varieties, J. Algebr. Geom., Volume 8 (1999) no. 2, pp. 197-206 | MR | Zbl

[44] Mukai, Shigeru Biregular classification of Fano 3-folds and Fano manifolds of coindex 3, Proc. Natl. Acad. Sci. USA, Volume 86 (1989) no. 9, pp. 3000-3002 | DOI | MR | Zbl

[45] Peternell, Thomas; Schrack, Florian Contact Kähler manifolds: symmetries and deformations, Algebraic and complex geometry (Springer Proceedings in Mathematics & Statistics), Volume 71, Springer, 2014, pp. 285-308 | DOI | MR | Zbl

[46] Sernesi, Edoardo Deformations of algebraic schemes, Grundlehren der Mathematischen Wissenschaften, 334, Springer, 2006, xii+339 pages | MR

[47] Stacks Project Authors Stacks Project, http://stacks.math.columbia.edu, 2017

[48] Ye, Yun-Gang A note on complex projective threefolds admitting holomorphic contact structures, Invent. Math., Volume 115 (1994) no. 2, pp. 311-314 | MR | Zbl

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