×

Injective endomorphisms of algebraic and analytic sets. (English) Zbl 0761.14005

The authors prove that every injective endomorphism of an algebraic variety over an algebraically closed field of characteristic zero is an automorphism. This result can be considered as a geometric counterpart of Grothendieck’s form of the Zariski main theorem. Such result generalizes a well-known result of A. Białynicki-Birula and M. Rosenlicht [Proc. Am. Math. Soc. 13, 200–203 (1962; Zbl 0107.14602)] saying that every injective polynomial transformation of \(\mathbb{R}^n\) is a polynomial automorphism. Moreover, the surjectivity of an injective endomorphism of an affine algebraic variety was earlier proved by J. Ax (using the “transfer principle” and the metamathematical notion of an “elementary formula”). A topological proof of the Ax theorem was given by A. Borel in an unpublished paper.
In the last section of the paper under review, the authors give a counterexample in the category of analytic sets, constructing a holomorphic homeomorphic bijection \(F: V\to V\) of an irreducible analytic curve \(V\subset\mathbb{C}^6\) onto itself with \(F^{-1}\) non-holomorphic.

MSC:

14H37 Automorphisms of curves
14E05 Rational and birational maps
32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables

Citations:

Zbl 0107.14602
PDFBibTeX XMLCite
Full Text: DOI