Injective endomorphisms of algebraic and analytic sets

Sławomir Cynk; Kamil Rusek

Annales Polonici Mathematici (1991)

  • Volume: 56, Issue: 1, page 29-35
  • ISSN: 0066-2216

Abstract

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We prove that every injective endomorphism of an affine algebraic variety over an algebraically closed field of characteristic zero is an automorphism. We also construct an analytic curve in ℂ⁶ and its holomorphic bijection which is not a biholomorphism.

How to cite

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Sławomir Cynk, and Kamil Rusek. "Injective endomorphisms of algebraic and analytic sets." Annales Polonici Mathematici 56.1 (1991): 29-35. <http://eudml.org/doc/262495>.

@article{SławomirCynk1991,
abstract = {We prove that every injective endomorphism of an affine algebraic variety over an algebraically closed field of characteristic zero is an automorphism. We also construct an analytic curve in ℂ⁶ and its holomorphic bijection which is not a biholomorphism.},
author = {Sławomir Cynk, Kamil Rusek},
journal = {Annales Polonici Mathematici},
keywords = {injective endomorphism; Zariski main theorem; polynomial automorphism},
language = {eng},
number = {1},
pages = {29-35},
title = {Injective endomorphisms of algebraic and analytic sets},
url = {http://eudml.org/doc/262495},
volume = {56},
year = {1991},
}

TY - JOUR
AU - Sławomir Cynk
AU - Kamil Rusek
TI - Injective endomorphisms of algebraic and analytic sets
JO - Annales Polonici Mathematici
PY - 1991
VL - 56
IS - 1
SP - 29
EP - 35
AB - We prove that every injective endomorphism of an affine algebraic variety over an algebraically closed field of characteristic zero is an automorphism. We also construct an analytic curve in ℂ⁶ and its holomorphic bijection which is not a biholomorphism.
LA - eng
KW - injective endomorphism; Zariski main theorem; polynomial automorphism
UR - http://eudml.org/doc/262495
ER -

References

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  1. [1] J. Ax, A metamathematical approach to some problems in number theory,, in: Proc. Sympos. Pure Math. 20, Amer. Math. Soc., 1971, 161-190. Zbl0218.10075
  2. [2] A. Białynicki-Birula and M. Rosenlicht, Injective morphisms of real algebraic varieties, Proc. Amer. Math. Soc. 13 (1962), 200-203. Zbl0107.14602
  3. [3] A. Borel, Injective endomorphisms of algebraic varieties, preprint. Zbl0189.21402
  4. [4] J. Dieudonné, Cours de géométrie algébrique, Vol. II, Presses Univ. France, 1974. 
  5. [5] A. Grothendieck, Eléments de géométrie algébrique. IV. Etude locale des schémas et des morphismes de schémas (quatrième partie), Inst. Hautes Etudes Sci. Publ. Math. 32 (1967). Zbl0153.22301
  6. [6] R. C. Gunning and H. Rossi, Analytic Functions of Several Complex Variables, Prentice-Hall, 1965. Zbl0141.08601
  7. [7] S. Łojasiewicz, An Introduction to Complex Analytic Geometry, PWN, Warszawa 1988 (in Polish). Zbl0773.32007
  8. [8] H. Matsumura and P. Monsky, On the automorphisms of hypersurfaces, J. Math. Kyoto Univ. 3 (3) (1964), 347-361. Zbl0141.37401
  9. [9] J.-P. Serre, Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier (Grenoble) 6 (1955-56), 1-42. 

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