Geometric invariant theory. Over the real and complex numbers
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 two essential results. The first is a very useful version of the Kempf-Ness theorem which is resumed here to show the style and strength of the book: Assume \(G\) is a closed real symmetric subgroup of \(\text{GL}(n,\mathbb R)\). Let \(K=G\cap \text{O}(n)\), \(\mathfrak{g}=\text{Lie}(G)\), \(\mathfrak{k}=\text{Lie}(K)\), \(\mathfrak{p}=\mathfrak{k}^\perp\cap \mathfrak{g}\), \(\mathfrak{a}\) a Cartan subspace of \(\mathfrak{p}\), and \(A=\exp(\mathfrak a)\). Let \(\langle\dots,\dots\rangle\) denote the standard inner product on \(\mathbb R^n\), let \(v\in\mathbb R^n.\) 1. \(v\) is critical if and only if \(\parallel gv\parallel\geq \parallel v\parallel\) for all \(g\in G\). 2. If \(v\) is critical, and \(X\in\mathfrak p\) is such that \(\parallel e^Xv\parallel=\parallel v\parallel\), then \(Xv=0.\) If \(w\in Gv\) is such that \(\parallel v\parallel=\parallel w\parallel\), then \(w\in Kv\). 3. If \(Gv\) is closed, then there exists a critical element in \(Gv\). The second result is the conjugacy result due to Cartan, Malcev and Mostow: Let \(G\) be a symmetric real subgroup of \(\mathrm{GL}(n,\mathbb R)\) and let \(K_1\) be a compact subgroup of \(G\). Then there exists \(X\in\mathfrak{p}=\{X\in\mathrm{Lie}(G)|X^T=X\}\) such that \(e^X K_1e^{-X}\subset K=\mathrm{O}(n)\cap G.\) After the background chapter, the book goes into the geometric invariant theory, in a much more explicit way than other textbooks. This is possible, working over \(\mathbb C\) or \(\mathbb R\). The affine basics of GIT is nicely introduced, that is, closed orbits and representations. This text even include the definition of a linearly reductive group, a group with the property that every regular representation is completely reducible, and the text contains proofs of several results not easy found in other textbooks. Such as for instance that a symmetric subgroup of \(\mathrm{GL}(n,\mathbb C)\) is linearly reductive. The book gives a nice definition of the Reynolds operators and a proof of Hilbert's finiteness theorem for invariants: If \(G\) is linearly reductive and \(X\) affine, then \(\mathcal O(X)^G\) is finitely generated over \(\mathbb C\). By this, the definition of a categorical quotient is more easy. With the background of the Matsushima's theorem, it follows that if \(G\) is a linearly reductive subgroup of \(\mathrm{GL}(n,\mathbb C)\), then there exists \(g\in\mathrm{GL}(n,\mathbb C)\) such that \(gGg^{-1}\) is a symmetric subgroup. Until this point, the second part can be introduced in the algebraic theory, not involving much of the Euclidean geometry. This changes when it comes to homogeneous spaces. The action of a Lie group is now studied as differentiable manifolds, and this is proved to agree with the algebraic definition of Lie algebra actions. This claims the definition of algebraic homogeneous spaces, and eventually gives a stringent proof of the fact that for \(X\) a variety, \(G\) an affine algebraic group acting on \(X\) and \(x\in X\), the obvious homomorphism \(G\rightarrow G_x\) induces an isomorphism \(G/G_x\simeq G_x.\) From this point on, the main topic of the book starts evolving. The usual invariants and closed orbits for those are introduced. Both the Euclidean and the algebraic theory is needed to prove the Hilbert-Mumford theorem, this again leading to the Kemp-Ness theorem over \(\mathbb C\). Giving explicit examples, this leads to an interpretation of the \(S\)-topology of a categorical quotient. At this point, everything that can be done directly, for subgroups of \(\mathrm{GL}(n,\mathbb C)\) has been stated and proved. To get any further, (without deformation theory/ theory of moduli), the category has to be extended. This is Vinbergs theory which is given in the rest of the third chapter in the second part. A Vinberg pair is a pair \((H,V)\) of a finite dimensional \(\mathbb C\)-vector space \(V\) and a Zariski closed, connected, reductive subgroup of \(\mathrm{GL}(V)\) satisfying the following conditions: 1. There exists a reductive Lie algebra \(\mathfrak g\) over \(\mathbb C\) and an automorphism \(\theta\) of \(\mathfrak g\) of order \(m\). 2. There is a primitive \(m\)-th root of unity \(\zeta\), such that \(V\) is the \(\zeta\) eigenspace for \(\theta\) and \(\mathrm{Lie}(H)=\mathfrak g^\theta\). This is what is called \(\theta\)-groups by Vinberg, and the following exposition follows his original ideas. More recent results by Panyushev are included, and these are used to prove Vinberg's main theorem without case-by-case consideration. One special case with \(m>2\) is considered. This case relies on Springer's theory of regular elements in reflection groups, and the results of Springer give a complete proof of Vinberg's theorem in these cases. This section also includes complete proofs of the Shepard-Todd theorem and a generalization of the Kostant-Rallis multiplicity theorem for Vinberg pairs. The technical part of the section includes gradings of Lie algebras, Vinberg triples, decomposition into simple components, the null cone, Cartan subspaces, the Weyl group and invariants of a regular Vinberg pair. Following the exposition of Vinberg pairs are several explicit examples and an appendix on root systems and their application. Section 4 studies weight theory in GIT: From the basics about solvable groups, Borel fixed point theorem, Borel subgroups, via regular representations of reductive groups, roots and weights, to Kostant's quadratic generation theorem, the Casimir operator and Kostant's formulas. The section ends with explicit interesting examples. The final section, section 5, consider classical and geometric invariant theory for products of classical groups and applies the Cartan-Helgason Theorem in the classification. This leads to examples interesting for physics, such as qubits and \(n\)-qudits, and even bra and ket-notation. The book is written as a graduate course in classical and geometric invariant theory. The first part gives a nice introduction to the subject that can be understood on the background of algebraic geometry and differential geometry, and the interplay between the two fields is of great value. The second part proves the classical results based on the developed theory, and makes the course close to complete over \(\mathbb C\) and \(\mathbb R\).
scientific article; zbMATH DE number 6764645
scientific article