Endomorphisms of homogeneous manifolds
type: Item
instance of: scholarly article
title: Endomorphisms of homogeneous manifolds
zbMATH Open document ID: 1059.32003
author: Serge Cantat
published in: L'Enseignement Mathématique. 2e Série
publication date: 2004-10-25T00:00:00Z
review text: Let \(X\) be a compact complex manifold. An endomorphism of \(X\) is a holomorphic map of \(X\) onto itself. The author studies endomorphisms that have topological degree greater than one, particularly in the case that \(X\) is homogeneous. An important notion for this study is the concept of an endomorphism that admits an invertible factor. Suppose \(\dim X>0\) and there is a locally trivial holomorphic fiber bundle \(\pi:X\to B\), where \(\dim B>0\), and the endomorphism \(f\) of \(X\) permutes the fibers of \(\pi\). Then \(f\) is said to admit an invertible factor, if the induced map on \(B\) is an automorphism, i.e., the induced endomorphism on the ``factor space'' \(B\) is invertible. The main result that the author proves is the following. Suppose \(X\) is a compact, complex, homogeneous manifold and \(f\) is an automorphism of \(X\) that does not have an invertible factor. Then there exists a locally trivial holomorphic fibration (in fact, a homogeneous fibration) \(\pi:X\to Q = {\mathbb P}_{m_{1}} \times\ldots\times {\mathbb P}_{m_{k}}\), whose fiber is a nilmanifold. Let \(f_{Q}\) denote the induced endomorphism of \(Q\). There is a positive integer \(l\) such that \(f_{Q}^{l}\) is a diagonal map \((f_{1}, \ldots,f_{k})\), where for each \(1 \leq i \leq k\) the map \(f_{i}: {\mathbb P}_{m_{i}} \to {\mathbb P}_{m_{i}}\) is an endomorphism that is not injective. The ingredients in the proof include the normalizer fibration [or \textit{J. Tits}' fibration, see, e.g., Comment. Math. Helv. 37, 111--120 (1962; Zbl 0108.36302) or \textit{D. N. Akhiezer's} book `Lie Group Actions in Complex Analysis', Braunschweig: Vieweg (1995; Zbl 0845.22001)] that shows that \(X\) fibers over a flag manifold with fiber a complex parallelizable manifold, a theorem of \textit{K.-H. Paranjape} and \textit{V. Srinivas} [Invent. Math. 98, 425--444 (1998; Zbl 0697.14037)] that deals with endomorphisms of flag manifolds, used to draw the conclusion about the base of the normalizer fibr…
reviewed by: Bruce Gilligan
Mathematics Subject Classification ID: 32M10
Mathematics Subject Classification ID: 32Q15
zbMATH DE Number: 2109717
zbMATH Keywords: homogeneous complex manifolds
zbMATH Keywords: compact Kähler manifolds
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2109717