Holomorphic maps onto varieties of non-negative Kodaira dimension

Authors:
Jun-Muk Hwang, Stefan Kebekus and Thomas Peternell

Journal:
J. Algebraic Geom. **15** (2006), 551-561

DOI:
https://doi.org/10.1090/S1056-3911-05-00411-X

Published electronically:
April 21, 2005

MathSciNet review:
2219848

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Abstract | References | Additional Information

Abstract: A classical result in complex geometry says that the automorphism group of a manifold of general type is discrete. It is more generally true that there are only finitely many surjective morphisms between two fixed projective manifolds of general type. Rigidity of surjective morphisms, and the failure of a morphism to be rigid have been studied by a number of authors in the past. The main result of this paper states that surjective morphisms are always rigid, unless there is a clear geometric reason for it. More precisely, we can say the following.

First, deformations of surjective morphisms between normal projective varieties are unobstructed unless the target variety is covered by rational curves.

Second, if the target is not covered by rational curves, then surjective morphisms are infinitesimally rigid, unless the morphism factors via a variety with positive-dimensional automorphism group. In this case, the Hom-scheme can be completely described.

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Additional Information

**Jun-Muk Hwang**

Affiliation:
Korea Institute for Advanced Study, 207-43 Cheongnyangni-dong, Seoul, 130-722, Korea

Email:
jmhwang@kias.re.kr

**Stefan Kebekus**

Affiliation:
Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Köln, Germany

Email:
stefan.kebekus@math.uni-koeln.de

**Thomas Peternell**

Affiliation:
Institut für Mathematik, Universität Bayreuth, 95440 Bayreuth, Germany

Email:
thomas.peternell@uni-bayreuth.de

DOI:
https://doi.org/10.1090/S1056-3911-05-00411-X

Received by editor(s):
January 21, 2005

Received by editor(s) in revised form:
February 24, 2005

Published electronically:
April 21, 2005

Additional Notes:
Jun-Muk Hwang was supported by the Korea Research Foundation Grant (KRF-2002-070-C00003). Stefan Kebekus was supported by a Heisenberg-Fellowship of the DFG. Jun-Muk Hwang, Stefan Kebekus and Thomas Peternell were supported in part by the program “Globale Methoden in der Komplexen Analysis” of the DFG