Deformation of holomorphic maps onto Fano manifolds of second and fourth Betti numbers 1
Annales de l'Institut Fourier, Volume 57 (2007) no. 3, p. 815-823
Let X be a Fano manifold with b 2 =1 different from the projective space such that any two surfaces in X have proportional fundamental classes in H 4 (X,C). Let f:YX be a surjective holomorphic map from a projective variety Y. We show that all deformations of f with Y and X fixed, come from automorphisms of X. The proof is obtained by studying the geometry of the integral varieties of the multi-valued foliation defined by the variety of minimal rational tangents of X.
Soit X une variété de Fano avec b 2 =1 différente de l’espace projectif et telle que tout couple de surfaces dans X ont des classes fondamentales dans H 4 (X,C) proportionnelles. Soit f:YX une application surjective d’une variété projective Y dans X. Nous montrons que toute déformation de f de Y dans X (fixés), provient d’automorphismes de X. La preuve est obtenue en étudiant la géométrie des variétés intégrales du feuilletage multi-valué défini par la variété des vecteurs tangents des courbes rationnelles minimales de X.
DOI : https://doi.org/10.5802/aif.2278
Classification:  14J45,  32H02
Keywords: minimal rational curves, Fano manifold, deformation of holomorphic maps
@article{AIF_2007__57_3_815_0,
     author = {Hwang, Jun-Muk},
     title = {Deformation of holomorphic maps onto Fano manifolds  of second and fourth Betti numbers 1},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {57},
     number = {3},
     year = {2007},
     pages = {815-823},
     doi = {10.5802/aif.2278},
     zbl = {1126.32011},
     mrnumber = {2336831},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2007__57_3_815_0}
}
Hwang, Jun-Muk. Deformation of holomorphic maps onto Fano manifolds  of second and fourth Betti numbers 1. Annales de l'Institut Fourier, Volume 57 (2007) no. 3, pp. 815-823. doi : 10.5802/aif.2278. https://aif.centre-mersenne.org/item/AIF_2007__57_3_815_0/

[1] Amerik, E. On a problem of Noether-Lefschetz type, Compositio Mathematica, Tome 112 (1998), pp. 255-271 | Article | MR 1631767 | Zbl 0929.14003

[2] Araujo, C. Rational curves of minimal degree and characterization of projective spaces, Math. Annalen, Tome 335 (2006), pp. 937-951 | Article | MR 2232023 | Zbl 05046898

[3] Hwang, J.-M. Geometry of minimal rational curves on Fano manifolds, ICTP Lect. Notes, Abdus Salam Int. Cent. Theoret. Phys.,, Trieste, School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000), Tome 6 (2001), pp. 335-393 | MR 1919462 | Zbl 01816818

[4] Hwang, J.-M. On the degrees of Fano four-folds of Picard number 1, J. reine angew. Math., Tome 556 (2003), pp. 225-235 | Article | MR 1971147 | Zbl 1016.14022

[5] Hwang, J.-M.; Kebekus, S.; Peternell, T. Holomorphic maps onto varieties of non-negative Kodaira dimension, J. Alg. Geom., Tome 15 (2006), pp. 551-561 | Article | MR 2219848 | Zbl 05135140

[6] Hwang, J.-M.; Mok, N. Finite morphisms onto Fano manifolds of Picard number 1 which have rational curves with trivial normal bundles, J. Alg. Geom., Tome 12 (2003), pp. 627-651 | Article | MR 1993759 | Zbl 1038.14018

[7] Hwang, J.-M.; Mok, N. Birationality of the tangent map for minimal rational curves, Asian J. Math., Tome 8 (2004), pp. 51-64 (Special issue dedicated to Yum-Tong Siu) | MR 2128297 | Zbl 1072.14015

[8] Okonek, C.; Schneider, M.; Spindler, H. Vector bundles on complex projective spaces, Birkhäuser, Boston (1980) | MR 561910 | Zbl 0438.32016