Discrete groups of reflections in the complex ball (Q1069007)

To give a signature on a complex manifold one specifies an open cover of the manifold by simply-connected subsets \(U_{\alpha}\) together with surjective analytic maps \(\pi_{\alpha}: \tilde U_{\alpha}\to U_{\alpha '}\) where each \(\tilde U_{\alpha}\) is an open neighbourhood of zero in \({\mathbb{C}}^ n\) invariant under the action of a finite group \(\Gamma_{\alpha}\) of automorphisms of \({\mathbb{C}}^ n\), and the fibres of \(\pi_{\alpha}\) are orbits of \(\Gamma_{\alpha}\). This paper is concerned with the question whether, given a signature S on a compact connected algebraic manifold X, there exists a Galois covering \(\pi: Y\to X\) with Y projective space, affine space or the complex ball such that S is the signature induced by \(\pi\). Some necessary conditions and examples are given.