Varieties of minimal rational tangents of codimension 1

Hwang, Jun-Muk

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 46 (2013) no. 4, pp. 629-649.

Soit $X$ une variété projective uniréglée et soit $x$ un point général. D’après le résultat principal de [2], si le degré par rapport à $-K_X$ de toute courbe rationnelle passant par $x$ est au moins égal à $dim\left(X\right)+1$, alors $X$ est un espace projectif. Dans cet article, nous étudions la structure de $X$ sous l’hypothèse que le degré par rapport à $-K_X$ de toute courbe rationnelle passant par $x$ est au moins égal à $dim\left(X\right)$. Notre étude repose sur la variété projective ${𝒞}_x\subset \mathbb{P}{T}_x\left(X\right)$ que nous appelons la VMRT (variété des tangentes des courbes rationnelles minimales) en $x$ et qui est définie comme la réunion de toutes les directions tangentes aux courbes rationnelles passant par $x$ dont le degré par rapport à $-K_X$ est minimal. Lorsque ce degré est égal à $dim\left(X\right)$, la VMRT ${𝒞}_x$ est une hypersurface de $\mathbb{P}{T}_x\left(X\right)$. Notre résultat principal affirme que si la VMRT en un point général d’une variété projective uniréglée $X$ de dimension $\ge 4$ est une hypersurface, alors $X$ est birationnelle au quotient d’une variété rationnelle explicite par l’action d’un groupe fini. Si, de plus, le rang du groupe de Picard de $X$ est égal à $1$, nous en déduisons que $X$ est une hypersurface quadrique d’un espace projectif.

Let $X$ be a uniruled projective manifold and let $x$ be a general point. The main result of [2] says that if the $(-K_X)$-degrees (i.e., the degrees with respect to the anti-canonical bundle of $X$) of all rational curves through $x$ are at least $dimX+1$, then $X$ is a projective space. In this paper, we study the structure of $X$ when the $(-K_X)$-degrees of all rational curves through $x$ are at least $dimX$. Our study uses the projective variety ${𝒞}_x\subset \mathbb{P}{T}_x\left(X\right)$, called the VMRT at $x$, defined as the union of tangent directions to the rational curves through $x$ with minimal $(-K_X)$-degree. When the minimal $(-K_X)$-degree of rational curves through $x$ is equal to $dimX$, the VMRT ${𝒞}_x$ is a hypersurface in $\mathbb{P}{T}_x\left(X\right)$. Our main result says that if the VMRT at a general point of a uniruled projective manifold $X$ of dimension $\ge 4$ is a smooth hypersurface, then $X$ is birational to the quotient of an explicit rational variety by a finite group action. As an application, we show that, if furthermore $X$ has Picard number 1, then $X$ is biregular to a hyperquadric.

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

[2] K. Cho, Y. Miyaoka & N. I. Shepherd-Barron, Characterizations of projective space and applications to complex symplectic manifolds, in Higher dimensional birational geometry (Kyoto, 1997), Adv. Stud. Pure Math. 35, Math. Soc. Japan, 2002, 1-88. | MR

[3] J. Hong & J.-M. Hwang, Characterization of the rational homogeneous space associated to a long simple root by its variety of minimal rational tangents, in Algebraic geometry in East Asia-Hanoi 2005, Adv. Stud. Pure Math. 50, Math. Soc. Japan, 2008, 217-236. | 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

[5] J.-M. Hwang, Equivalence problem for minimal rational curves with isotrivial varieties of minimal rational tangents, Ann. Sci. Éc. Norm. Supér. 43 (2010), 607-620. | Numdam | MR

[6] J.-M. Hwang & N. Mok, Holomorphic maps from rational homogeneous spaces of Picard number $1$ onto projective manifolds, Invent. Math. 136 (1999), 209-231. | MR

[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

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

[9] S. Kebekus, Families of singular rational curves, J. Algebraic Geom. 11 (2002), 245-256. | MR

[10] S. Kobayashi & T. Ochiai, Characterizations of complex projective spaces and hyperquadrics, J. Math. Kyoto Univ. 13 (1973), 31-47. | MR

[11] J. Kollár, Rational curves on algebraic varieties, Ergebn. Math. Grenzg. 32, Springer, 1996.

[12] Y. Miyaoka, Numerical characterisations of hyperquadrics, in Complex analysis in several variables-Memorial Conference of Kiyoshi Oka's Centennial Birthday, Adv. Stud. Pure Math. 42, Math. Soc. Japan, 2004, 209-235. | MR

[13] 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. Parts 1, 2, AMS/IP Stud. Adv. Math. 42, Amer. Math. Soc., 2008, 41-61. | MR

[14] C. Okonek, M. Schneider & H. Spindler, Vector bundles on complex projective spaces, Progress in Math. 3, Birkhäuser, 1980. | MR

[15] K. Yamaguchi, Differential systems associated with simple graded Lie algebras, in Progress in differential geometry, Adv. Stud. Pure Math. 22, Math. Soc. Japan, 1993, 413-494. | MR