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

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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.

[Am]Am Amerik, E., *Maps onto certain Fano threefolds. Documenta Mathematica* **2** (1997) 195-211.
[ARV]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]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]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]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]IS Iliev, A. and Schuhmann, C., *Tangent scrolls in prime Fano threefolds*. Kodai Math. J. **23** (2000) 411-431.
[Is]Is Iskovskikh, V.A., *Anticanonical models of 3-dimensional algebraic varieties*. J. Soviet Math. **13** (1980) 745-814.
[Kd]Kd Kodaira, K., *On stability of compact submanifolds of complex manifolds*. Amer. J. Math. **85** (1963) 79-94.
[Kl]Kl Kollár, J., *Rational curves on algebraic varieties.* Ergebnisse der Mathematik und ihrer Grenzgebiete, 3 Folge, Band 32, Springer Verlag, 1996.
[KO]KO Kobayashi, S. and Ochiai, T., *Meromorphic mappings onto compact complex spaces of general type*. Invent. math. **31** (1975) 7-16.
[Ma]Ma Maehara, K., *A finiteness property of varieties of general type*. Math. Ann. **262** (1983) 101-123.
[MU]MU Mukai, S. and Umemura, H., *Minimal rational threefolds*. in *Algebraic Geometry, Tokyo/Kyoto 1982*. Lecture notes in Math. **1016** (1983) 490-518.
[Sc]Sc Schuhmann, C., *Morphisms between Fano threefolds* J. Alg. Geom. **8** (1999) 221-244
[YY]YY Yau, Stephen S.-T. and Yu, Y., *Gorenstein quotient singularities in dimension three*, Memoirs AMS **105** (1993).

[Am]Am Amerik, E., *Maps onto certain Fano threefolds. Documenta Mathematica* **2** (1997) 195-211.
[ARV]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]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]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]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]IS Iliev, A. and Schuhmann, C., *Tangent scrolls in prime Fano threefolds*. Kodai Math. J. **23** (2000) 411-431.
[Is]Is Iskovskikh, V.A., *Anticanonical models of 3-dimensional algebraic varieties*. J. Soviet Math. **13** (1980) 745-814.
[Kd]Kd Kodaira, K., *On stability of compact submanifolds of complex manifolds*. Amer. J. Math. **85** (1963) 79-94.
[Kl]Kl Kollár, J., *Rational curves on algebraic varieties.* Ergebnisse der Mathematik und ihrer Grenzgebiete, 3 Folge, Band 32, Springer Verlag, 1996.
[KO]KO Kobayashi, S. and Ochiai, T., *Meromorphic mappings onto compact complex spaces of general type*. Invent. math. **31** (1975) 7-16.
[Ma]Ma Maehara, K., *A finiteness property of varieties of general type*. Math. Ann. **262** (1983) 101-123.
[MU]MU Mukai, S. and Umemura, H., *Minimal rational threefolds*. in *Algebraic Geometry, Tokyo/Kyoto 1982*. Lecture notes in Math. **1016** (1983) 490-518.
[Sc]Sc Schuhmann, C., *Morphisms between Fano threefolds* J. Alg. Geom. **8** (1999) 221-244
[YY]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

MR Author ID:
362260

Email:
jmhwang@ns.kias.re.

**Ngaiming Mok**

Affiliation:
Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong

MR Author ID:
191186

Email:
nmok@hkucc.hku.hk

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