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Geometric invariant theory and decorated principal bundles. (English) Zbl 1159.14001
Zurich Lectures in Advanced Mathematics. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-065-4/pbk). vii, 389 p. (2008).
Generally speaking, the book under review deals with classification problems in algebraic geometry from the viewpoint of geometric invariant theory (GIT). It consists of two major parts, both of which differ essentially in their purpose, size, specification, topicality, and expository character, respectively.
Part 1 occupies less than one third of the text and provides a profound introduction to the basic concepts, methods, and results of modern geometric invariant theory as developed by D. Mumford and his successors in the 1960s and thereafter. Being much less sophisticated and technical than Mumford’s famous original research monograph [D. Mumford, J. Fogarty and F. Kirwan “Geometric Invariant Theory” Berlin: Springer (1993; Zbl 0797.14004)] from 1965, this introductory part is based on the author’s repeated lectures on the subject, delivered at the University of Zürich (1996/1997 and 2006) and at the University of Barcelona (1997) a few years ago. Accordingly, Part 1 has nearly textbook character, and a good acquaintance with the very basic theory of algebraic varieties should suffice as a solid background for the access to it by non-specialists.
Part 2 is more in the style of a research monograph and presents the author’s recent work on special classes of algebraic principal fibre bundles and their moduli spaces from the aspect of geometric invariant theory. Although the reader will need here some familiarity with schemes, sheaves, Serre’s GAGA principle, and the concept of moduli spaces, most of the necessary prerequisites are also thoroughly recalled in the course of the text, thereby making the latter overall fairly self-contained and widely accessible for seasoned students and researchers in complex geometry. As for the more precise contents of the book, these can be summarized as follows:
Part 1, the introduction to basic geometric, invariant theory for complex algebraic varieties, encompasses seven sections, each of which is subdivided into several subsections. Section 1 discusses linear algebraic groups and their (homogeneous) representations, reductive groups, and the reductivity of the classical groups via H. Weyl’s “unitarian trick”. Section 2 introduces geometric invariant theory for affine varieties on the level of vector spaces and representations, whereas Section 3 gives the standard examples from the classical invariant theory of algebraic forms and matrices. Section 4 turns to the basic concepts of Mumford’s geometric invariant theory, including the various kinds of quotients modulo group actions. Criteria for stability and semi-stability are the subject of Section 5, ranging from the classical Hilbert-Mumford criterion up to some recent refined semi-stability criteria due to the author himself [cf.: A. H. W. Schmitt, Proc. Am. Math. Soc. 129, No. 7, 1923–1926 (2001; Zbl 1002.14015)] together with many concrete examples. Section 6 deals with linearizations and the variation of quotients à la Dolgachev-Hu, Bialynicki-Birula, Thaddeus, and the author himself. Section 7 addresses some further recent results concerning specific refinements of the Hilbert-Mumford criterion of stability, most of which have not been included in the earlier relevant textbooks so far, but which are crucial for many constructions in Part 2 of the book. The main novelty of the author’s exposition is the elementary discussion of the finiteness of the number of different quotients for the same group action and the variation of the quotients [cf.: A. H. W. Schmitt, Proc. Am. Math. Soc. 131, No. 2, 359–362 (2003; Zbl 1008.14009)] given in the foregoing Section 6, together with the large number of instructive examples in each section of Part 1.
Part 2, the more advanced research part and the core of the book, is devoted to the classification problem for a particular class of algebro-geometric objects, the so-called “decorated principal bundles” over an algebraic curve.
Section 1 gives the necessary account of the theory of algebraic fibre spaces (with fibre \(F\) and structure group \(G)\) and principal \(G\)-bundles over a variety \(X\), following the masterly classic exposition by J.-P. Serre in Séminaire Chevelley 1958. Also, in this introductory section, the fundamental classification problem underlying the entire second part of the book is precisely formulated. Section 2 briefly reviews the basic material concerning the classification of algebraic vector bundles over a curve \(X\), thereby demonstrating the specific use of geometric invariant theory for constructing moduli spaces, on the one hand, and providing the necessary background material for the later sections on the other. Section 3 describes decorated vector bundles over curves and the solution to the general classification problem in this special case. The results presented here are due to the author himself [Transform. Groups 9, No. 2, 167–209 (2004; Zbl 1092.14042)], culminating in a universal construction for moduli spaces of decorated vector bundles over a curve, which extends the related previous work by C. Simpson [Publ. Math., Inst. Hautes Étud. Sci. 79, 47–129 (1994; Zbl 0891.14005)] and by D. Huybrechts and M. Lehn [Int. J. Math. 6, 297–324 (1995; Zbl 0865.14004); J. Algebr. Geom. 4, 67–104 (1995; Zbl 0839.14023)]. Section 4 offers one of the highlights of the present monograph, namely a full construction of the moduli space of semi-stable principal \(G\)-bundles with connected reductive structure group. Using the results on decorated vector bundles, this provides a new approach to that particular problem, which was first tackled by A. Ramanathan more than ten years ago [cf.: A. Ramanathan, Indian Acad. Sci., Math. Sci. 106, No. 3 and 4, 301–328 and 421–449 (1996; Zbl 0901.14007 and Zbl 0901.14008)]. The author’s exposition presented here is also the first one occurring sytematically in a monograph, therefore representing a particularly valuable general reference with respect to this topic. Actually, the material originates from the author’s recent research reports on the subject published in 2002 and 2004 [cf.: A. H. W. Schmitt, Int. Math. Res. Not. 2002, No. 23, 1183–1209 (2002; Zbl 1034.14017; and 3327–3366 (2004; Zbl 1093.14507)]. In three appendices, the author has added some complementing remarks to this section. Section 5 investigates decorated tuples of vector bundles and their moduli spaces. The structure group is here a finite product of general linear groups over \(\mathbb{C}\), and the associated protective moduli problem comprises several new and important aspects. The construction of the respective moduli spaces is carried out in great detail, including all the necessary tricks and technicalities, among which are representation spaces for quivers [cf.: A. H. W. Schmitt, Proc. Indian Acad. Sci. Math. Sci. 115, 15–49 (2005; Zbl 0076.14019)].
Section 6 is devoted to an extension of the formalism for principal \(G\)-bundles developed in Section 4. The methods established here allow to handle also the case of an arbitrary, not necessarily connected reductive structure group \(G\) as well as the more general, so-called “pseudo \(G\)-bundles” in the sequel. Section 7 contains the absolute novelties of the present monograph which consist in a far-reaching generalization of the results of Sections 3 and 5. This leads to a complete proof of the main theorem on the existence and the geometric structure of moduli spaces for principal \(G\)-bundles decorated with sections in an associated protective bundle. To this end, the author refines the notion of semi-stability for the objects under consideration in an appropriate manner, and he even generalizes his classification result to (suitably defined) decorated pseudo \(G\)-bundles. Next, the asymptotic behaviour of the semi-stability concept is carefully analyzed, followed by an instructive application of the author’s general results. Namely, it is explained how the compactified moduli space of semi-stable Higgs bundles can be recovered from the moduli theory for principal \(G\)-bundles as a special case.
Section 8 finally presents the ultimate main result of the monograph under review as announced in Section 1. In fact, this result establishes the existence of the moduli space of (properly defined) semi-stable principal \(G\)-bundles which are decorated by a section in the vector bundle associated with a fixed representation of the structure group \(G\). This is achieved by linking this moduli problem to the constructions carried out in the previous sections, which turns out to be rather tricky and involved. The significance of the author’s general moduli construction becomes evident from the examples and applications presented in the sequel, including another model for the moduli space of Bradlow pairs as well as moduli spaces of Higgs bundles for real reductive groups. The final Section 9 gives an outlook to potential extensions and further generalizations of the concepts, methods, and results presented in this monograph. These concern (1) the case of positive characteristic of the ground field, (2) the case of higher-dimensional base varieties, and (3) decorated parabolic bundles, where things naturally become more complicated. However, the author discusses the present state of art, in this respect, and he suggests some useful ideas for further research in these directions.
The book comes with a rich up-to-date bibliography (222 references) and with an utmost carefully compiled index. The entire exposition stands out by its exemplary lucidity, comprehensiveness, profundity, depth, and rigor. No doubt, both students and researchers in algebraic geometry can benefit enormously from this masterly and topical monograph.

MSC:
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14L24 Geometric invariant theory
14L30 Group actions on varieties or schemes (quotients)
14H60 Vector bundles on curves and their moduli
14D20 Algebraic moduli problems, moduli of vector bundles
13-02 Research exposition (monographs, survey articles) pertaining to commutative algebra
13A50 Actions of groups on commutative rings; invariant theory
20G05 Representation theory for linear algebraic groups
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