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Abelian covers of algebraic varieties. (English) Zbl 0721.14009

The paper outlines a theory of normal abelian covers of smooth algebraic varieties, generalizing the well known analysis of double covers and of some special instances of cyclic covers. - Given an abelian cover \(\pi\) : \(X\to Y\) with group G, as above, one defines the “building data” of such a cover to be the eigensheaves of \(\pi_*{\mathcal O}_ X\) corresponding to the different characters of G and certain subsets of the branch divisor of \(\pi\). The main result is a necessary and sufficient condition for the existence of a cover with prescribed building data.
Moreover the paper contains an analysis of the geometric properties of X (singularities and so on...) in terms of the properties of Y and of the building data, the computation of the direct images via \(\pi\) of the tangent, cotangent and canonical sheaves of X and a definition of “natural deformations” of an abelian cover.
Reviewer: R.Pardini (Pisa)

MSC:

14E20 Coverings in algebraic geometry
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