Compact complex surfaces admitting non-trivial surjective endomorphisms
type: Item
instance of: scholarly article
title: Compact complex surfaces admitting non-trivial surjective endomorphisms
zbMATH Open document ID: 1088.32008
author: Yoshio Fujimoto
author: Noboru Nakayama
published in: Tôhoku Mathematical Journal. Second Series
publication date: 2006-01-26T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.tmj/1128703004
review text: In the paper under review a non-trivial surjective endomorphism of a connected compact complex manifold \(X\) is understood as a surjective holomorphic map \(X\rightarrow X\) which is not biholomorphic. For dim \(X=1\) the Riemann-Hurwitz formula shows that \(X\) admits a non-trivial endomorphism if and only if it is isomorphic to the projective line. But in dimension two the determination of all \(X\) which admit non-trivial surjective endomorphisms is much more difficult. The projective algebraic case was solved by former results of both authors [see \textit{Y. Fujimoto}, Publ. Res. Inst. Math. Sci. 38, No. 1, 33--92 (2002; Zbl 1053.14049) and \textit{N. Nakayama}, Kyushu J. Math. 56, No. 2, 433--446 (2002; Zbl 1049.14029)]. The list consists of toric surfaces, \(\mathbb P^1\)-bundles over elliptic curves and specific \(\mathbb P^1\)-bundles over hyperbolic curves. In the present article they solve the non-algebraic case, thereby completing the two-dimensional list. The non-algebraic list consists of complex tori (algebraic dimension \(a(X)\leq 1\), \(b_1(X)\) even), the primary and secondary Kodaira surfaces and the elliptic Hopf surfaces (\(b_1(X)\) odd, \(a(X)=1\)), Hopf surfaces with two elliptic curves and specific Inoue surfaces of class \(VII_0\) (\(a(X)=0,b_1(X)=1, b_2(X)=0\)), and specific Inoue surfaces of class \(VII\). The authors prove that a compact complex surface \(X\) with a non-trivial surjective endomorphism contains only finitely many complex curves with negative self-intersection. This fact has strong implications. Case by case considerations and the use of the Enriques-Kodaira classification of compact complex surfaces yield the final list.
reviewed by: Eberhard Oeljeklaus
Mathematics Subject Classification ID: 32J15
Mathematics Subject Classification ID: 14J25
Mathematics Subject Classification ID: 14J27
zbMATH DE Number: 5001917
zbMATH Keywords: surjective holomorphic endomorphism
zbMATH Keywords: Kodaira surface
zbMATH Keywords: Inoue surface
MaRDI profile type: Publication
OpenAlex ID: W2065245849
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DOI: 10.2748/TMJ/1128703004
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description: scientific article; zbMATH DE number 5001917