Invariant Hypersurfaces in holomorphic Dynamics

We prove the following result, which is analogous to two theorems, one due to Kodaira and Krasnov and another one due to Jouanolou and Ghys. Let $M$ be a compact complex manifold and $f$ a dominant endomorphism of $M.$ If there exist $k$ totally invariant irreducible hypersurfaces $W_i\subset M,$ with $k > \dim(M)+ h^{1,1}(M)$ then $f$ preserves a nontrivial meromorphic fibration. We then study the case where $f$ is a meromorphic map.

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Published 1 January 2010