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Endomorphisms of smooth projective 3-folds with non-negative Kodaira dimension. (English) Zbl 1053.14049
From the introduction: The main purpose of this paper is to study the structure of a nonsingular projective 3-fold $$X$$ with a surjective morphism $$f:X\to X$$ onto itself which is not an isomorphism, called a nontrivial surjective endomorphism of $$X$$. Let $$f:X\to X$$ be a surjective morphism from a nonsingular projective variety $$X$$ onto itself. Then $$f$$ is a finite morphism and if the Kodaira dimension $$\kappa(X)$$ of $$X$$ is non-negative, $$f$$ is a finite étale covering. The structure of an algebraic surface $$S$$ which admits a nontrivial surjective endomorphism is fairly simple. If $$\kappa(S)\geq 0$$, $$S$$ is minimal and a suitable finite étale covering of $$S$$ is isomorphic to an abelian surface or the direct product of an elliptic curve and a smooth curve of genus $$\geq 2$$.
Let $$X$$ be a smooth projective 3-fold with $$\kappa(X)=0$$ or 2 which admits a nontrivial surjective endomorphism $$f:X\to X$$. The author shows that a suitable finite étale covering $$\widetilde X$$ of $$X$$ has the structure of a smooth abelian scheme over a nonsingular projective variety $$W$$ with $$0\leq\dim (W)<\dim(X)$$. Moreover, $$\widetilde X$$ can be chosen to be isomorphic to an abelian 3-fold or the direct product $$E\times W$$ of an elliptic curve $$E$$ and a smooth projectice surface $$W$$ with $$\kappa(W)=\kappa(X)$$.

##### MSC:
 14J30 $$3$$-folds 14J15 Moduli, classification: analytic theory; relations with modular forms 14E20 Coverings in algebraic geometry 14E30 Minimal model program (Mori theory, extremal rays)
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