Equivalence problem for minimal rational curves with isotrivial varieties of minimal rational tangents

Hwang, Jun-Muk

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 43 (2010) no. 4, pp. 607-620.

Nous énonçons le problème d’équivalence, au sens de É. Cartan, pour des familles de courbes rationnelles minimales sur des variétés projectives uniréglées. Un invariant important de ce problème d’équivalence est la variété des tangentes rationnelles minimales. Nous étudions le cas où les variétés de tangentes rationnelles minimales aux points génériques forment une famille isotriviale. La question principale dans ce cas est  : pour quelle variété projective $Z$ une famille de courbes rationnelles minimales, dont les variétés de tangentes rationnelles minimales sont $Z$-isotriviales, est-elle localement équivalente au modèle plat  ? Nous montrons que c’est le cas lorsque $Z$ vérifie certaines conditions de géométrie projective qui sont satisfaites pour une hypersurface non singulière de degré $\ge 4$.

We formulate the equivalence problem, in the sense of É. Cartan, for families of minimal rational curves on uniruled projective manifolds. An important invariant of this equivalence problem is the variety of minimal rational tangents. We study the case when varieties of minimal rational tangents at general points form an isotrivial family. The main question in this case is for which projective variety $Z$, a family of minimal rational curves with $Z$-isotrivial varieties of minimal rational tangents is locally equivalent to the flat model. We show that this is the case when $Z$ satisfies certain projective-geometric conditions, which hold for a non-singular hypersurface of degree $\ge 4$.

[1] C. Araujo, Rational curves of minimal degree and characterizations of projective spaces, Math. Ann. 335 (2006), 937-951. | MR | Zbl

[2] P. Brückmann & H.-G. Rackwitz, $T$-symmetrical tensor forms on complete intersections, Math. Ann. 288 (1990), 627-635. | MR | Zbl

[3] É. Cartan, Les sous-groupes des groupes continus de transformations, Ann. Sci. École Norm. Sup. 25 (1908), 57-194. | JFM | Numdam | MR

[4] J.-M. Hwang, Geometry of minimal rational curves on Fano manifolds, in School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000), ICTP Lect. Notes 6, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001, 335-393. | MR | Zbl

[5] J.-M. Hwang & N. Mok, Uniruled projective manifolds with irreducible reductive $G$-structures, J. reine angew. Math. 490 (1997), 55-64. | MR | Zbl

[6] J.-M. Hwang & N. Mok, Varieties of minimal rational tangents on uniruled projective manifolds, in Several complex variables (Berkeley, CA, 1995-1996), Math. Sci. Res. Inst. Publ. 37, Cambridge Univ. Press, 1999, 351-389. | MR | Zbl

[7] J.-M. Hwang & N. Mok, Cartan-Fubini type extension of holomorphic maps for Fano manifolds of Picard number 1, J. Math. Pures Appl. 80 (2001), 563-575. | MR | Zbl

[8] J.-M. Hwang & N. Mok, Birationality of the tangent map for minimal rational curves, Asian J. Math. 8 (2004), 51-63. | MR | Zbl

[9] J.-M. Hwang & N. Mok, Prolongations of infinitesimal linear automorphisms of projective varieties and rigidity of rational homogeneous spaces of Picard number 1 under Kähler deformation, Invent. Math. 160 (2005), 591-645. | MR | Zbl

[10] N. Mok, Recognizing certain rational homogeneous manifolds of Picard number 1 from their varieties of minimal rational tangents, in Third International Congress of Chinese Mathematicians. Part 1, 2, AMS/IP Stud. Adv. Math., 42, pt. 1 2, Amer. Math. Soc., 2008, 41-61. | MR | Zbl

[11] S. Sternberg, Lectures on differential geometry, second éd., Chelsea Publishing Co., 1983. | MR | Zbl