Cyclic and abelian coverings of real varieties
The authors study the birational and biregular theory of cyclic and, more generally, abelian coverings of real algebraic varieties, the latter being regarded as complex varieties equipped with an anti-holomorphic involution \(\sigma\). They outline the difficulty of the problem, due to the fact that \(\sigma\) is only required to normalize the Galois group, not necessarily commute with it. In the cyclic birational case (field extensions), this results in four different models (Theorem 3.2) instead of the familiar single complex equation \(z^n=f\). The cyclic biregular theory is summarized in Theorem 4.2, which is a relatively straightforward real refinement of the complex results, in terms of the ramification divisor and certain additional linear systems, by [\textit{R. Pardini}, J. Reine Angew. Math. 417, 191--213 (1991; Zbl 0721.14009)]. Unfortunately, the principal statements are too long to be cited in this review. As an important preliminary result, the authors also state the real version, in terms of the so-called \textit{real fundamental group}, of the Riemann existence theorem.
scientific article; zbMATH DE number 7710888
scientific article