Cyclic and abelian coverings of real varieties
type: Item
instance of: scholarly article
title: Cyclic and abelian coverings of real varieties
zbMATH Open document ID: 1523.14099
arXiv ID: 2108.03754
author: Fabrizio Catanese
author: Michael Lönne
author: Fabio Perroni
published in: Mathematische Annalen
publication date: 2023-07-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/2108.03754
review text: 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.
reviewed by: Alex Degtyarev
Mathematics Subject Classification ID: 14P99
Mathematics Subject Classification ID: 14P25
Mathematics Subject Classification ID: 14H30
Mathematics Subject Classification ID: 30F50
Mathematics Subject Classification ID: 12D99
zbMATH DE Number: 7710888
zbMATH Keywords: real algebraic variety
zbMATH Keywords: cyclic covering
zbMATH Keywords: abelian covering
MaRDI profile type: Publication
OpenAlex ID: W3189957427
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DOI: 10.1007/S00208-022-02439-Z
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description: scientific article; zbMATH DE number 7710888