Injective endomorphisms of algebraic varieties (Q5906603)

The main purpose of this paper is to present the following theorem on injective endomorphisms of algebraic varieties: Consider an algebraic variety \(X\) over an algebraically closed field \(k\) of characteristic zero (i.e. \(X\) is a reduced separated \(k\)-scheme of finite type). Then every injective endomorphism \(f:X \to X\) is necessarily an automorphism of \(X\). This theorem generalizes certain results of \textit{J. Ax} [Ann. Math., II. Ser. 88, 239-271 (1968; Zbl 0195.057)], \textit{A. Borel} [Arch. Math. 20, 531-537 (1969; Zbl 0189.214)]as well as of \textit{S. Cynk} and \textit{K. Rusek} [Ann. Pol. Math. 56, No. 1, 29-35 (1991; Zbl 0761.14005)]. The proof given here is based on Grothendieck's theory of projective systems of schemes and on Zariski's main theorem on birational transformations.