Geometric invariant theory. Over the real and complex numbers
type: Item
instance of: scholarly article
title: Geometric invariant theory. Over the real and complex numbers
zbMATH Open document ID: 1387.14124
DOI: 10.1007/978-3-319-65907-7
author: Nolan Wallach
published in: Universitext
publication date: 2017-08-25T00:00:00Z
review text: This is a textbook in geometric invariant theory with basis in Mumfords geometric invariant theory (GIT), over the complex and real numbers. As algebraic geometry, and in particular GIT, gets its geometric ideas from Euclidean, or rather differential geometry, the link between the Zariski topology and the ordinary Euclidean geometry is essential. The book starts with a part on background theory. The first chapter in this part consider algebraic geometry with a functorial definition and view toward the comparison between the algebraic (Zariski) and the geometric (Euclidean) topology. The chapter is a very intense and concentrated review, and shows how geometric methods can be used algebaically and vice versa. In addition, the theory of algebraic tangent spaces of projective and quasi-projective varieties are taught in a very natural way. As the author states in the introduction to this chapter, highlights are complete proofs of the difficult fact that a smooth algebraic variety over \(\mathbb C\) is a differentiable manifold in the metric topology, and the fact that the closure of a Zariski-open set in the metric topology is the same as its Zariski closure. The later chapters on GIT depends heavily on these results. Chapter 2 of the first part is a module on Lie groups and algebraic groups. Again, these objects are related by their common definitions for different topologies. The closure of subgroups, and the existence of maximal compact subgroups, are essential for later applications. In particular, the symmetric subgroups and their decompositions are introduced. The classical correspondence \(G\mapsto\text{Lie}(G)\) between Lie groups and Lie algebras is given, and explained in an intuitive way. By this, the classification of the symmetric subgroups of \(\text{GL}(n,\mathbb R)\) shows its value, and the task is performed. Then using (the classical canonical) weight-theory, the compact and the algebraic tori are defined and compared. The background part ends with tw…
reviewed by: Arvid Siqveland
Mathematics Subject Classification ID: 14L24
Mathematics Subject Classification ID: 14L05
Mathematics Subject Classification ID: 13A50
zbMATH DE Number: 6764645
zbMATH Keywords: geometric invariant theory
zbMATH Keywords: GIT
zbMATH Keywords: Lie group
zbMATH Keywords: Lie algebra
zbMATH Keywords: tangent spaces
zbMATH Keywords: representation theory
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-319-65907-7
OpenAlex ID: W4213298845
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description: scientific article; zbMATH DE number 6764645