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
DOI:
https://doi.org/10.1090/S1056391103003199
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.

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Additional Information
JunMuk Hwang
Affiliation:
Korea Institute for Advanced Study, 20743 Cheongryangridong, Seoul 130012, 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. 980701015L from the KOSEF. Supported by a grant of the Hong Kong Research Grants Council