Compact varieties of surjective holomorphic endomorphisms
Let X denote a normal compact connected reduced complex space. By Douady, the set \({\mathcal H}(X,X)\) of all holomorphic mappings \(X\to X,\) with the compact-open topology, carries a complex structure. Denote by \(Sur X\) a subset of \({\mathcal H}(X,X)\), which consists of all holomorphic surjections \(X\to X.\) The author shows that \(Sur X\) is open and closed in \({\mathcal H}(X,X)\), in particular it is analytic. The main result of the paper says that if Z is any compact connected reduced subspace of \(Sur X,\) then for every two surjections \(\sigma\), \(\tau\in Z\), there exists a biholomorphic automorphism \(\alpha\) of X such that \(\sigma =\alpha \circ \tau\). Such an \(\alpha\) can be described more exactly. An example shows, that the normality of X is essential. The author uses standard methods of complex analytic geometry and complex Lie groups.
scientific article; zbMATH DE number 3929422