Étale endomorphisms of 3-folds. I.
type: Item
instance of: scholarly article
title: Étale endomorphisms of 3-folds. I.
zbMATH Open document ID: 1394.14025
author: Yoshio Fujimoto
published in: Osaka Journal of Mathematics
publication date: 2018-05-18T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ojm/1524038727
review text: Let \(X\) be a smooth projective variety of dimension 3 defined over \(\mathbb{C}\). The purpose of this paper is to apply minimal model program (MMP) to develop a strategy for classifications of threefolds admitting a non-isomorphic étale endomorphism (an endomorphism is a morphism from the variety to itself) in case of \(\kappa(X)=-\infty\). The infinite sequence \(Z_\cdot =(v_n:Z_n\rightarrow Z_{n+1})_{n\in Z}\) is an étale sequence of constant Picard number (ESP for short) if the following three conditions are satisfied (1) any \(Z_n\) is a smooth projective variety, (2) any \(v_n\) is a non-isomorphic finite étale covering, and (3) the Picard number \(\rho(Z_n)=\dim_{\mathbb{R}}N_1(Z_n)\) is a constant. An ESP \(Y_{\cdot}=(g_n: Y_n\rightarrow Y_{n+1})\) is an ESP of fiber type (FESP for short) if any \(Y_n\) \((n\in \mathbb{Z})\) is a smooth projective 3-fold with an extremal ray of fiber type. Recall the notations in birational geometry. \(N_1(X)\) is the \({\mathbb{R}}\)-vector space of 1-cycles with real coefficients modulo numerical equivalence. \(\mathrm{NE}(X)= \{ \sum a_i[C_i]: C_i\subset X, 0\leq a_i\in {\mathbb{R}} \}\subset N_1(X) \), \(\overline{\mathrm{NE}}(X)=\) the closure of \(\mathrm{NE}(X)\) in \(N_1(X)\), where the \(C_i\) are irreducible curves on \(X\). Let \(V\) be a vector space over \(\mathbb{R}\). A subset \(N\subset V\) is a cone if \(0\in N\) and \(N\) is closed under multiplication by positive scalars. A subcone \(M\subset N\) is extremal (or an extremal face of \(N\)) if \(x, y\in N\), \(x+y\in M\) imply \(x, y\in M\). A 1-dimensional extremal subcone is called an extremal ray. In this paper, an extremal ray \(R\) on a smooth projective variety \(X\) means a \(1-\)dimensional face of \(\overline{\mathrm{NE}}(X)\) with \(K_XR<0\). The motivation of this paper is to apply MMP to classify smooth projective 3-folds \(X\) with \(\kappa(X)=-\infty\) admitting a non-isomorphic étale endomorphism. The author introduces the notions of an ESP …
reviewed by: Jing Zhang
Mathematics Subject Classification ID: 14J15
Mathematics Subject Classification ID: 14J25
Mathematics Subject Classification ID: 14G35
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 14J60
Mathematics Subject Classification ID: 32J17
Mathematics Subject Classification ID: 14E30
Mathematics Subject Classification ID: 14J50
zbMATH DE Number: 6870388
zbMATH Keywords: étale endomorphism
zbMATH Keywords: MMP
zbMATH Keywords: 3-folds
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6870388