Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

Finite morphisms onto Fano manifolds of Picard number 1 which have rational curves with trivial normal bundles


Authors: Jun-Muk Hwang and Ngaiming Mok
Journal: J. Algebraic Geom. 12 (2003), 627-651
DOI: https://doi.org/10.1090/S1056-3911-03-00319-9
Published electronically: April 10, 2003
MathSciNet review: 1993759
Full-text PDF

Abstract | References | Additional Information

Abstract: Let $X$ be a Fano manifold of Picard number 1 admitting a rational curve with trivial normal bundle and $f\colon X'\to X$ be a generically finite surjective holomorphic map from a projective manifold $X'$ onto $X$. When the domain manifold $X'$ is fixed and the target manifold $X$ is a priori allowed to deform we prove that the holomorphic map $f\colon X'\to X$ is locally rigid up to biholomorphisms of target manifolds. This result complements, with a completely different method of proof, an earlier local rigidity theorem of ours (see J. Math. Pures Appl. 80 (2001), 563-575) for the analogous situation where the target manifold $X$ is a Fano manifold of Picard number $1$on which there is no rational curve with trivial normal bundle. In another direction, given a Fano manifold $X'$ of Picard number $1$, we prove a finiteness result for generically finite surjective holomorphic maps of $X'$onto Fano manifolds (necessarily of Picard number $1$) admitting rational curves with trivial normal bundles. As a consequence, any $3$-dimensional Fano manifold of Picard number $1$ can only dominate a finite number of isomorphism classes of projective manifolds.


References [Enhancements On Off] (What's this?)

  • [Am] Amerik, E., Maps onto certain Fano threefolds. Documenta Mathematica 2 (1997) 195-211.
  • [ARV] Amerik, E., Rovinsky, M. and Van de Ven, A., A boundedness theorem for morphisms between threefolds, Annal. L'Institut Fourier 49 (1999) 405-415.
  • [HM1] Hwang, J.-M. and Mok, N., Holomorphic maps from rational homogeneous spaces of Picard number 1 onto projective manifolds. Invent. math. 136 (1999) 209-231.
  • [HM2] Hwang, J.-M. and Mok, N., Varieties of minimal rational tangents on uniruled manifolds. in Several Complex Variables, ed. by M. Schneider and Y.-T. Siu, MSRI Publications 37, Cambridge University Press (2000) 351-389.
  • [HM3] Hwang, J.-M. and Mok, N., Cartan-Fubini type extension of holomorphic maps for Fano manifolds of Picard number 1, J. Math. Pures Appl. 80 (2001) 563-575.
  • [IS] Iliev, A. and Schuhmann, C., Tangent scrolls in prime Fano threefolds. Kodai Math. J. 23 (2000) 411-431.
  • [Is] Iskovskikh, V.A., Anticanonical models of 3-dimensional algebraic varieties. J. Soviet Math. 13 (1980) 745-814.
  • [Kd] Kodaira, K., On stability of compact submanifolds of complex manifolds. Amer. J. Math. 85 (1963) 79-94.
  • [Kl] Kollár, J., Rational curves on algebraic varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3 Folge, Band 32, Springer Verlag, 1996.
  • [KO] Kobayashi, S. and Ochiai, T., Meromorphic mappings onto compact complex spaces of general type. Invent. math. 31 (1975) 7-16.
  • [Ma] Maehara, K., A finiteness property of varieties of general type. Math. Ann. 262 (1983) 101-123.
  • [MU] Mukai, S. and Umemura, H., Minimal rational threefolds. in Algebraic Geometry, Tokyo/Kyoto 1982. Lecture notes in Math. 1016 (1983) 490-518.
  • [Sc] Schuhmann, C., Morphisms between Fano threefolds J. Alg. Geom. 8 (1999) 221-244
  • [YY] Yau, Stephen S.-T. and Yu, Y., Gorenstein quotient singularities in dimension three, Memoirs AMS 105 (1993).


Additional Information

Jun-Muk Hwang
Affiliation: Korea Institute for Advanced Study, 207-43 Cheongryangri-dong, Seoul 130-012, Korea
Email: jmhwang@ns.kias.re.

Ngaiming Mok
Affiliation: Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong
Email: nmok@hkucc.hku.hk

DOI: https://doi.org/10.1090/S1056-3911-03-00319-9
Received by editor(s): December 18, 2000
Published electronically: April 10, 2003
Additional Notes: Supported by Grant No. 98-0701-01-5-L from the KOSEF. Supported by a grant of the Hong Kong Research Grants Council

American Mathematical Society