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 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), 563-575) 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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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