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April 2018 Étale endomorphisms of 3-folds. I
Yoshio Fujimoto
Osaka J. Math. 55(2): 195-257 (April 2018).

Abstract

This paper is the first part of our project towards classifications of smooth projective $3$-folds $X$ with $\kappa(X) = -\infty$ admitting a non-isomorphic étale endomorphism. We can prove that for any extremal ray $R$ of divisorial type, the contraction morphism $\pi_R\colon X\to X'$ associated to $R$ is the blowing-up of a smooth $3$-fold $X'$ along an elliptic curve. The difficulty is that there may exist infinitely many extremal rays on $X$. Thus we introduce the notion of an `ESP' which is an infinite sequence of non-isomorphic finite étale coverings of $3$-folds with constant Picard number. We can run the minimal model program (`MMP') with respect to an ESP and obtain the `FESP' $Y_{\bullet}$ of $(X, f)$ which is a distinguished ESP with \textit{extremal rays of fiber type} (cf. Definition 3.6). We first classify $Y_{\bullet}$ and then blow-up $Y_{\bullet}$ along elliptic curves to recover the original $X$. The finiteness of extremal rays of $\overline{\rm NE}(X)$ is verified in certain cases (cf. Theorem 1.4). We encounter a new phenomenon showing that our \'{e}taleness assumption is related with torsion line bundles on an elliptic curve (cf. Theorem 1.5).

Citation

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Yoshio Fujimoto. "Étale endomorphisms of 3-folds. I." Osaka J. Math. 55 (2) 195 - 257, April 2018.

Information

Published: April 2018
First available in Project Euclid: 18 April 2018

zbMATH: 06870388
MathSciNet: MR3787744

Subjects:
Primary: 14J15, 14J25, 14J30, 14J60
Secondary: 32J17

Rights: Copyright © 2018 Osaka University and Osaka City University, Departments of Mathematics

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Vol.55 • No. 2 • April 2018
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