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 \(\kappa(W)=\kappa(X)\).
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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