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

Author(s): Jun-Muk Hwang; Ngaiming Mok
Journal: J. Algebraic Geom. 12 (2003), 627-651.
Posted: April 10, 2003
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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.

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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

PII: S 1056-3911(03)00319-9
Received by editor(s): December 18, 2000
Posted: 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

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