On a cyclic covering of a projective manifold (Q802680)

The author proves here some statements concerning the structure of an abelian cover f: \(Y\to X\), with X smooth and Y normal. He proves that if the branch divisor B of f is irreducible and the cover is totally ramified, then f is a simple cyclic cover, namely it can be embedded in the total space of a line bundle L such that rL is linearly equivalent to B, r being the degree of the cover. Specializing to the case in which X is an abelian variety and Y is of general type the author computes the cohomology of the structure sheaf on Y. - Finally he studies the singularities of a cyclic cover of \({\mathbb{P}}^ 2\) branched on the union of two smooth curves. For more general results of the same type see also ``Abelian covers of algebraic varieties'' by the reviewer [J. Reine Angew. Math. 417, 191-213 (1991; Zbl 0721.14009)].