Cylinders in Fano varieties
type: Item
instance of: scholarly article
title: Cylinders in Fano varieties
zbMATH Open document ID: 1492.14016
DOI: 10.4171/EMSS/44
author: Ji-Hun Park
author: Mikhail Zaidenberg
author: I. A. Cheltsov
author: Yuri Prokhorov
published in: EMS Surveys in Mathematical Sciences
publication date: 2021-09-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/2007.14207
review text: A \textit{cylinder} in a complex projective variety \(X\) is a Zariski open subset \(U\) of \(X\) isomorphic to a product \(Z\times\mathbb{C}\) for some affine variety \(Z\). The article contains a survey of the current knowledge concerning the existence and non-existence of cylinders in mildly singular del Pezzo surfaces, in higher dimensional smooth Fano varieties and in total spaces of certain other Mori fiber spaces over positive dimensional bases. A particular emphasis is put on the question of existence of cylinders with respect to a given fixed ample polarization \(H\) on \(X\), the so-called \textit{\(H\)-polar cylinders}, which are cylinders \(U\) in \(X\) whose complements are the supports of effective \(\mathbb{Q}\)-divisors \(\mathbb{Q}\)-linearly equivalent to \(H\). A very natural case of special importance in the Fano context is that of anti-canonically polar cylinders. The article also reviews several inspiring connections to other classical problems such as rationality and \(K\)-stability of Fano varieties, existence and properties of compactification of the affine space \(\mathbb{C}^{n}\) into smooth and mildly singular Fano varieties and existence and properties of additive group actions on affine cones over polarized projective varieties.
reviewed by: Adrien Dubouloz
Mathematics Subject Classification ID: 14E05
Mathematics Subject Classification ID: 14J45
Mathematics Subject Classification ID: 14J50
Mathematics Subject Classification ID: 14R20
Mathematics Subject Classification ID: 14R25
Mathematics Subject Classification ID: 14E08
Mathematics Subject Classification ID: 14E30
Mathematics Subject Classification ID: 14-02
zbMATH DE Number: 7394433
zbMATH Keywords: cylinder
zbMATH Keywords: Fano variety
zbMATH Keywords: unipotent group action
MaRDI profile type: Publication
arXiv ID: 2007.14207
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