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  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
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.


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

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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: http://dx.doi.org/10.1090/S1056-3911-03-00319-9
PII: S 1056-3911(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


Journal of Algebraic Geometry
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