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Holomorphic maps onto varieties of non-negative Kodaira dimension. (English) Zbl 1112.14014
Summary: 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 numher 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 hy 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.

##### MSC:
 14E07 Birational automorphisms, Cremona group and generalizations 14A10 Varieties and morphisms 14J50 Automorphisms of surfaces and higher-dimensional varieties 14J40 $$n$$-folds ($$n>4$$) 14E05 Rational and birational maps 32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
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##### References:
 [1] Armand Borel and Raghavan Narasimhan, Uniqueness conditions for certain holomorphic mappings, Invent. Math. 2 (1967), 247 – 255. · Zbl 0145.31802 [2] G. Dethloff and H. Grauert, Seminormal complex spaces, Several complex variables, VII, Encyclopaedia Math. Sci., vol. 74, Springer, Berlin, 1994, pp. 183 – 220. · Zbl 0832.32008 [3] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. · Zbl 0367.14001 [4] J.-M. Hwang. Rigidity of surjective holomorphic maps to Calabi-Yau manifolds. Math. Zeit. 249 (4): 767-772, 2005. · Zbl 1070.32018 [5] Shoshichi Kobayashi and Takushiro Ochiai, Meromorphic mappings onto compact complex spaces of general type, Invent. Math. 31 (1975), no. 1, 7 – 16. · Zbl 0331.32020 [6] Flips and abundance for algebraic threefolds, Société Mathématique de France, Paris, 1992. Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, August 1991; Astérisque No. 211 (1992) (1992). [7] János Kollár, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 32, Springer-Verlag, Berlin, 1996. · Zbl 0877.14012 [8] Morris Kalka, Bernard Shiffman, and Bun Wong, Finiteness and rigidity theorems for holomorphic mappings, Michigan Math. J. 28 (1981), no. 3, 289 – 295. · Zbl 0459.32011 [9] David I. Lieberman, Compactness of the Chow scheme: applications to automorphisms and deformations of Kähler manifolds, Fonctions de plusieurs variables complexes, III (Sém. François Norguet, 1975 – 1977) Lecture Notes in Math., vol. 670, Springer, Berlin, 1978, pp. 140 – 186. · Zbl 0391.32018 [10] Hideyuki Matsumura, On algebraic groups of birational transformations, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 34 (1963), 151 – 155. · Zbl 0134.16601 [11] Yoichi Miyaoka, The Chern classes and Kodaira dimension of a minimal variety, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 449 – 476. [12] V. B. Mehta and A. Ramanathan, Semistable sheaves on projective varieties and their restriction to curves, Math. Ann. 258 (1981/82), no. 3, 213 – 224. · Zbl 0473.14001 [13] Thomas Peternell and Andrew J. Sommese, Ample vector bundles and branched coverings, Comm. Algebra 28 (2000), no. 12, 5573 – 5599. With an appendix by Robert Lazarsfeld; Special issue in honor of Robin Hartshorne. · Zbl 0982.14025 [14] Thomas Peternell and Andrew J. Sommese, Ample vector bundles and branched coverings. II, The Fano Conference, Univ. Torino, Turin, 2004, pp. 625 – 645. · Zbl 1071.14018 [15] Maxwell Rosenlicht, Some basic theorems on algebraic groups, Amer. J. Math. 78 (1956), 401 – 443. · Zbl 0073.37601
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