On branched coverings of projective manifolds (Q1820214)

This paper consists of an announcement of a criterion for a divisor in a complex manifold to be the branch locus of a Galois covering. The criterion is a condition on the set of all isomorphism classes of unitary flat vector bundles with prescribed singularities. It is a difficult condition to implement, but leads to a more amenable criterion in the case where M is projective and the divisor consists of irreducible hypersurfaces with certain intersection properties, a result which generalizes a theorem of \textit{M. Kato} [Mem. Fac. Sci., Kyushu Univ., Ser. A 38, 127-131 (1984; Zbl 0556.32007)].