Stable reductive varieties. II: Projective case
type: Item
instance of: scholarly article
title: Stable reductive varieties. II: Projective case
zbMATH Open document ID: 1080.14017
DOI: 10.1016/S0001-8708(03)00164-6
author: Michel Brion
author: Valery Alexeev
published in: Advances in Mathematics
publication date: 2004-08-06T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0207274
review text: This is the second part of the authors work on stable reductive varieties where they extend the results of the affine part [Invent. Math. 157, No.~2, 227--274 (2004; Zbl 1068.14003)] to the projective setting. The authors construct a moduli space of stable reductive pairs which are the higher dimensional analog of stable \(n\)-pointed curves, and generalize to the noncommutative case a functorial compactification of moduli of abelian varieties, and of toric pairs. So, the main aim of the article, is to construct a compactified moduli of higher dimensional projective varieties in the special case when there is a nontrivial reductive group action. The compactification of the moduli of stable \(n\)-pointed curves \(\overline{M}_{g,n}\) was extended to pairs \((X,D)\) of log general type by Kollár, Shepherd-Barron, Alexeev, and others, using the methods of the log minimal program. Finally it was extended by Alexeev to the case of pairs \((X,D)\) in which the variety has arbitrary dimension but comes with a nontrivial semiabelian group action, for example, an abelian or toric variety. In the abelian case, this gives a moduli compactification of the moduli space of principally polarized abelian varieties. The toric case is closely related to \(A\)-hypergeometric functions. Each stable toric pair \((X,D)\) comes with a canonical map to \(\mathbb{P}^n\), and the image is its ``shadow cycle''. Now there is a toric Chow variety parameterizing these cycles. The authors study pairs of varieties with divisors rather than just polarized projective varieties. The most important reasons for this is the following: Without the additional structure there are only finitely many isomorphism classes of varieties of a fixed numerical type, the automorphism group is (possibly) infinite, stable limits does not necessarily exist uniquely. These things are repaired by adding the additional structure of the divisor. Also, the log minimal program states that varieties and pairs that have good m…
reviewed by: Arvid Siqveland
Mathematics Subject Classification ID: 14D22
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 14M25
Mathematics Subject Classification ID: 14A22
Mathematics Subject Classification ID: 14E30
zbMATH DE Number: 2081972
zbMATH Keywords: moduli
zbMATH Keywords: reductive groups
zbMATH Keywords: spherical varieties
zbMATH Keywords: minimal model program
zbMATH Keywords: log minimal program
zbMATH Keywords: stable reductive varieties
zbMATH Keywords: pairs
zbMATH Keywords: toric varieties
zbMATH Keywords: abelian varieties
MaRDI profile type: Publication
arXiv ID: math/0207274
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description: scientific article