Finite morphisms onto Fano manifolds of Picard number 1 which have rational curves with trivial normal bundles
Authors:
JunMuk Hwang and Ngaiming Mok
Journal:
J. Algebraic Geom. 12 (2003), 627651
Published electronically:
April 10, 2003
MathSciNet review:
1993759
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Abstract 
References 
Additional Information
Abstract: Let be a Fano manifold of Picard number 1 admitting a rational curve with trivial normal bundle and be a generically finite surjective holomorphic map from a projective manifold onto . When the domain manifold is fixed and the target manifold is a priori allowed to deform we prove that the holomorphic map 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), 563575) for the analogous situation where the target manifold is a Fano manifold of Picard number on which there is no rational curve with trivial normal bundle. In another direction, given a Fano manifold of Picard number , we prove a finiteness result for generically finite surjective holomorphic maps of onto Fano manifolds (necessarily of Picard number ) admitting rational curves with trivial normal bundles. As a consequence, any dimensional Fano manifold of Picard number can only dominate a finite number of isomorphism classes of projective manifolds.
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Hwang, J.M. and Mok, N., Holomorphic maps from rational homogeneous spaces of Picard number 1 onto projective manifolds. Invent. math. 136 (1999) 209231.
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Hwang, J.M. and Mok, N., CartanFubini type extension of holomorphic maps for Fano manifolds of Picard number 1, J. Math. Pures Appl. 80 (2001) 563575.
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Iliev, A. and Schuhmann, C., Tangent scrolls in prime Fano threefolds. Kodai Math. J. 23 (2000) 411431.
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Kollár, J., Rational curves on algebraic varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3 Folge, Band 32, Springer Verlag, 1996.
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Maehara, K., A finiteness property of varieties of general type. Math. Ann. 262 (1983) 101123.
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Mukai, S. and Umemura, H., Minimal rational threefolds. in Algebraic Geometry, Tokyo/Kyoto 1982. Lecture notes in Math. 1016 (1983) 490518.
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Schuhmann, C., Morphisms between Fano threefolds J. Alg. Geom. 8 (1999) 221244
 [YY]
Yau, Stephen S.T. and Yu, Y., Gorenstein quotient singularities in dimension three, Memoirs AMS 105 (1993).
 [Am]
 Amerik, E., Maps onto certain Fano threefolds. Documenta Mathematica 2 (1997) 195211.
 [ARV]
 Amerik, E., Rovinsky, M. and Van de Ven, A., A boundedness theorem for morphisms between threefolds, Annal. L'Institut Fourier 49 (1999) 405415.
 [HM1]
 Hwang, J.M. and Mok, N., Holomorphic maps from rational homogeneous spaces of Picard number 1 onto projective manifolds. Invent. math. 136 (1999) 209231.
 [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) 351389.
 [HM3]
 Hwang, J.M. and Mok, N., CartanFubini type extension of holomorphic maps for Fano manifolds of Picard number 1, J. Math. Pures Appl. 80 (2001) 563575.
 [IS]
 Iliev, A. and Schuhmann, C., Tangent scrolls in prime Fano threefolds. Kodai Math. J. 23 (2000) 411431.
 [Is]
 Iskovskikh, V.A., Anticanonical models of 3dimensional algebraic varieties. J. Soviet Math. 13 (1980) 745814.
 [Kd]
 Kodaira, K., On stability of compact submanifolds of complex manifolds. Amer. J. Math. 85 (1963) 7994.
 [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) 716.
 [Ma]
 Maehara, K., A finiteness property of varieties of general type. Math. Ann. 262 (1983) 101123.
 [MU]
 Mukai, S. and Umemura, H., Minimal rational threefolds. in Algebraic Geometry, Tokyo/Kyoto 1982. Lecture notes in Math. 1016 (1983) 490518.
 [Sc]
 Schuhmann, C., Morphisms between Fano threefolds J. Alg. Geom. 8 (1999) 221244
 [YY]
 Yau, Stephen S.T. and Yu, Y., Gorenstein quotient singularities in dimension three, Memoirs AMS 105 (1993).
Additional Information
JunMuk Hwang
Affiliation:
Korea Institute for Advanced Study, 20743 Cheongryangridong, Seoul 130012, 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:
http://dx.doi.org/10.1090/S1056391103003199
PII:
S 10563911(03)003199
Received by editor(s):
December 18, 2000
Published electronically:
April 10, 2003
Additional Notes:
Supported by Grant No. 980701015L from the KOSEF. Supported by a grant of the Hong Kong Research Grants Council
