Endomorphisms of homogeneous manifolds.
(Endomorphismes des variétés homogènes.)

*(French)*Zbl 1059.32003Let \(X\) 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 J. Tits’ fibration, see, e.g., Comment. Math. Helv. 37, 111–120 (1962; Zbl 0108.36302) or 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 K.-H. Paranjape and 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 fibration, and a result of J. Winkelmann [Transform. Groups 3, 103–111 (1998; Zbl 0902.32009)] on parallelizable manifolds, needed to understand the fiber of the normalizer fibration in the present setting. An additional key point is to understand how the isotropy subgroup of the flag manifold acts on the fiber of this fibration.

This paper also contains a fair amount of expository details, including a discussion of what happens in the case of compact, homogeneous Kähler manifolds and some instructive examples.

The last section contains an observation about compact Kähler manifolds (not necessarily homogeneous) that have nonnegative Kodaira dimension and a noninvertible endomorphism.

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 J. Tits’ fibration, see, e.g., Comment. Math. Helv. 37, 111–120 (1962; Zbl 0108.36302) or 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 K.-H. Paranjape and 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 fibration, and a result of J. Winkelmann [Transform. Groups 3, 103–111 (1998; Zbl 0902.32009)] on parallelizable manifolds, needed to understand the fiber of the normalizer fibration in the present setting. An additional key point is to understand how the isotropy subgroup of the flag manifold acts on the fiber of this fibration.

This paper also contains a fair amount of expository details, including a discussion of what happens in the case of compact, homogeneous Kähler manifolds and some instructive examples.

The last section contains an observation about compact Kähler manifolds (not necessarily homogeneous) that have nonnegative Kodaira dimension and a noninvertible endomorphism.

Reviewer: Bruce Gilligan (Regina)