##
**Geometric invariant theory.
3rd enl. ed.**
*(English)*
Zbl 0797.14004

Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge. 34. Berlin: Springer-Verlag. 320 p. (1994).

The first edition of D. Mumford’s celebrated “G.I.T.” (Geometric Invariant Theory) appeared in 1965 (Zbl 0147.393). At that time, it provided the first systematic and rigorous approach to the general algebraic fine-classification problem of algebro-geometric objects (over arbitrary base schemes). Building up on the rich work of the classical algebraic geometers of the Italian and German school, Weil’s and Zariski’s innovating ideas to renew algebraic geometry conceptually and methodically, and Grothendieck’s final rigorous foundation of algebraic geometry by consequently establishing the categorical and sheaf-theoretic framework, D. Mumford succeeded in formulating the classification problem for algebraic varieties and schemes properly, i.e., in terms of functors of families of algebro-geometric objects and their representability properties. Geometric invariant theory as the theory of orbit spaces of algebraic schemes acted on by algebraic groups, was developed as the crucial tool for tackling fine-classification problems (moduli problems) in algebraic geometry, and for investigating the geometric properties of the classifying schemes (moduli spaces), provided that their existence can be established. Apart from having set up the ultimate, rigorous framework for treating “moduli problems” in general, including the long-standing classical ones (e.g., for algebraic curves or compact Riemann surfaces, principally polarized abelian varieties, projective hypersurfaces of given degree, some classes of algebraic surfaces, etc.), Mumford gave – as a concrete application of his new moduli theory – the first complete construction of the moduli spaces for algebraic curves and polarized abelian varieties.

The first edition of Mumford’s “G.I.T.” was written as a research monograph, and his revolutionary ideas and results presented here, namely: the geometric formulation of classical invariant theory and its application to moduli problems in algebraic geometry, caused an explosion of activity in this old and new area of mathematics. During the following sixteen years, a tremendous progress took place, in both invariant theory and moduli theory, and Mumford’s “G.I.T.” undoubtedly served as the one and only standard text and reference book on these topics.

In 1982, when the reprinting of this monograph was already overdue, D. Mumford and his new co-author J. Fogarty took the opportunity of adding seven appendices to the (essentially unchanged) original text. In these appendices they sketched and commented on the progress relative to the topics treated in the original text, mostly without proofs, but with enlightening discussions, guiding hints and references to all the numerous research papers published in the meantime. In this way, the second enlarged edition of “G.I.T.” by Mumford and Fogarty became a sketchily updated version of the still brilliant original text (Zbl 0504.14008).

Now, after another twelve years, Mumford’s “G.I.T.”, together with the appendices of the second edition, is still maintaining its character as an unrivalled reference book on geometric invariant theory and moduli problems, and a third reprinting is certainly likewise desirable, justified and indispensable. What is more, in these twelve years there have been again so many developments relating to geometric invariant theory and moduli theory that another updating of both the comments in the appendices and the references became just as overdue. Indeed, the present third enlarged edition of “G.I.T.” comes up to these expectations. This time, the text was revised and complemented by a third co-author, namely F. Kirwan, and the changes compared to the second edition are as follows.

Firstly, F. Kirwan has added an extra chapter (chapter 8) to the seven chapters of the original text which, again, has been left intact. This new chapter, with the heading “The moment map”, takes account of one of the most significant developments, over the last decade, in geometric invariant theory and its cross links to complex differential geometry and mathematical physics. More precisely, appendix \(2C\) of the second edition already gave a brief hint at a (in those days) recent observation of D. Mumford [“Proof of the convexity theorem”], V. Guillemin and S. Sternberg [Invent. Math. 67, 491-513 (1982; Zbl 0503.58017) and 77, 533-546 (1984; Zbl 0561.58015)], that the concept of the moment map arising in symplectic geometry can be linked to the deep result of G. Kempf and L. Ness [in Algebraic geometry, Proc. Summer Meet., Copenhagen 1978, Lect. Notes Math. 732, 233-243 (1979; Zbl 0407.22012)] on characterizing stable vectors in complex representations of reductive groups. This relation between symplectic geometry and geometric invariant theory has been studied and exploited very extensively, in the meantime, and led to fruitful applications in mathematical physics (Yang-Mills theory). In chapter 8, F. Kirwan, who significantly contributed to this great progress, provides an overview of some of these developments. In nine sections she lucidly discusses the following topics:

1. Actions of Lie groups on symplectic manifolds and the moment map;

2. Symplectic quotients and geometric invariant theory;

3. Other quotient constructions: Kähler and hyperkähler quotients;

4. Singularities of symplectic quotients;

5. The geometry of the moment map and convexity properties;

6. The cohomology of quotients: the symplectic case;

7. The cohomology of quotients: the algebraic case;

8. Vector bundles and the Yang-Mills functional;

9. Yang-Mills theory over Riemann surfaces.

In style this new chapter is closer to the appendices added in the second edition of the book than to the original text. That means, in particular, no proofs are given where satisfactory references exist, but the philosophy, methods and main results are explained systematically, thoroughly, and in a very enlightening manner. The text is enhanced by numerous additional remarks given as footnotes, and by respective references to the (mostly very recent) original research papers and monographs. As for further reading and more detailed accounts, the reader should be referred, among many other treatises, to F. C. Kirwan’s monograph “Cohomology of quotients in symplectic and algebraic geometry”, Math. Notes 31 (1984; Zbl 0553.14020)], V. Guillemin’s recent book “Moment maps and combinatorial invariants of Hamiltonian \(T^ n\)-spaces”, Prog. Math. 122 (Basel 1994), and the paper by M. F. Atiyah and R. Bott [Philos. Trans. R. Soc. Lond. A. 308, 523- 615 (1983; Zbl 0509.14014)].

The second major enlargement of the new edition consists in an immense extension of the bibliography by 582 additional references. The updated list of references comprises 926 publications; this is about three times as much as in the second edition from twelve years ago, thereby indicating quantitatively what tremendous development the topic has undergone since then. In addition, F. Kirwan has amply supplied the appendices in the second edition – which otherwise have been left entirely unchanged – with numerous footnotes to each subsection. These footnotes mainly refer to recent work in the respective area as well as to the added references in the updated bibliography.

Altogether, the present third edition of Mumford’s “G.I.T.” has become something of an updated chronicle of the development of geometric invariant theory, moduli theory, and its (meanwhile) vast applications. It is actually a brilliant collage consisting of an evergreen, formerly epoch-making research monograph, a survey on a very recent topic (chapter 8), updatings of former updatings by appendices, and the perhaps most complete existing bibliography on this field of research, equipped with countlessly many guiding explanatory comments, hints, and inspiring invitations to independent research work. Of such a kind, the third edition of “G.I.T.” has corroborated the peerless character of this work as an indispensable standard reference book on invariant theory, moduli theory, and their recent applications in mathematical physics, which helps researchers and graduate students in these fields to keep up with the rapid developments and the vast literature.

The first edition of Mumford’s “G.I.T.” was written as a research monograph, and his revolutionary ideas and results presented here, namely: the geometric formulation of classical invariant theory and its application to moduli problems in algebraic geometry, caused an explosion of activity in this old and new area of mathematics. During the following sixteen years, a tremendous progress took place, in both invariant theory and moduli theory, and Mumford’s “G.I.T.” undoubtedly served as the one and only standard text and reference book on these topics.

In 1982, when the reprinting of this monograph was already overdue, D. Mumford and his new co-author J. Fogarty took the opportunity of adding seven appendices to the (essentially unchanged) original text. In these appendices they sketched and commented on the progress relative to the topics treated in the original text, mostly without proofs, but with enlightening discussions, guiding hints and references to all the numerous research papers published in the meantime. In this way, the second enlarged edition of “G.I.T.” by Mumford and Fogarty became a sketchily updated version of the still brilliant original text (Zbl 0504.14008).

Now, after another twelve years, Mumford’s “G.I.T.”, together with the appendices of the second edition, is still maintaining its character as an unrivalled reference book on geometric invariant theory and moduli problems, and a third reprinting is certainly likewise desirable, justified and indispensable. What is more, in these twelve years there have been again so many developments relating to geometric invariant theory and moduli theory that another updating of both the comments in the appendices and the references became just as overdue. Indeed, the present third enlarged edition of “G.I.T.” comes up to these expectations. This time, the text was revised and complemented by a third co-author, namely F. Kirwan, and the changes compared to the second edition are as follows.

Firstly, F. Kirwan has added an extra chapter (chapter 8) to the seven chapters of the original text which, again, has been left intact. This new chapter, with the heading “The moment map”, takes account of one of the most significant developments, over the last decade, in geometric invariant theory and its cross links to complex differential geometry and mathematical physics. More precisely, appendix \(2C\) of the second edition already gave a brief hint at a (in those days) recent observation of D. Mumford [“Proof of the convexity theorem”], V. Guillemin and S. Sternberg [Invent. Math. 67, 491-513 (1982; Zbl 0503.58017) and 77, 533-546 (1984; Zbl 0561.58015)], that the concept of the moment map arising in symplectic geometry can be linked to the deep result of G. Kempf and L. Ness [in Algebraic geometry, Proc. Summer Meet., Copenhagen 1978, Lect. Notes Math. 732, 233-243 (1979; Zbl 0407.22012)] on characterizing stable vectors in complex representations of reductive groups. This relation between symplectic geometry and geometric invariant theory has been studied and exploited very extensively, in the meantime, and led to fruitful applications in mathematical physics (Yang-Mills theory). In chapter 8, F. Kirwan, who significantly contributed to this great progress, provides an overview of some of these developments. In nine sections she lucidly discusses the following topics:

1. Actions of Lie groups on symplectic manifolds and the moment map;

2. Symplectic quotients and geometric invariant theory;

3. Other quotient constructions: Kähler and hyperkähler quotients;

4. Singularities of symplectic quotients;

5. The geometry of the moment map and convexity properties;

6. The cohomology of quotients: the symplectic case;

7. The cohomology of quotients: the algebraic case;

8. Vector bundles and the Yang-Mills functional;

9. Yang-Mills theory over Riemann surfaces.

In style this new chapter is closer to the appendices added in the second edition of the book than to the original text. That means, in particular, no proofs are given where satisfactory references exist, but the philosophy, methods and main results are explained systematically, thoroughly, and in a very enlightening manner. The text is enhanced by numerous additional remarks given as footnotes, and by respective references to the (mostly very recent) original research papers and monographs. As for further reading and more detailed accounts, the reader should be referred, among many other treatises, to F. C. Kirwan’s monograph “Cohomology of quotients in symplectic and algebraic geometry”, Math. Notes 31 (1984; Zbl 0553.14020)], V. Guillemin’s recent book “Moment maps and combinatorial invariants of Hamiltonian \(T^ n\)-spaces”, Prog. Math. 122 (Basel 1994), and the paper by M. F. Atiyah and R. Bott [Philos. Trans. R. Soc. Lond. A. 308, 523- 615 (1983; Zbl 0509.14014)].

The second major enlargement of the new edition consists in an immense extension of the bibliography by 582 additional references. The updated list of references comprises 926 publications; this is about three times as much as in the second edition from twelve years ago, thereby indicating quantitatively what tremendous development the topic has undergone since then. In addition, F. Kirwan has amply supplied the appendices in the second edition – which otherwise have been left entirely unchanged – with numerous footnotes to each subsection. These footnotes mainly refer to recent work in the respective area as well as to the added references in the updated bibliography.

Altogether, the present third edition of Mumford’s “G.I.T.” has become something of an updated chronicle of the development of geometric invariant theory, moduli theory, and its (meanwhile) vast applications. It is actually a brilliant collage consisting of an evergreen, formerly epoch-making research monograph, a survey on a very recent topic (chapter 8), updatings of former updatings by appendices, and the perhaps most complete existing bibliography on this field of research, equipped with countlessly many guiding explanatory comments, hints, and inspiring invitations to independent research work. Of such a kind, the third edition of “G.I.T.” has corroborated the peerless character of this work as an indispensable standard reference book on invariant theory, moduli theory, and their recent applications in mathematical physics, which helps researchers and graduate students in these fields to keep up with the rapid developments and the vast literature.

Reviewer: W.Kleinert (Berlin)

### MathOverflow Questions:

In GIT, why are the semistable/unstable loci defined pointwise, instead of defining semistable/unstable subschemes?### MSC:

14L24 | Geometric invariant theory |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

14D22 | Fine and coarse moduli spaces |

14L30 | Group actions on varieties or schemes (quotients) |

14D20 | Algebraic moduli problems, moduli of vector bundles |

14H10 | Families, moduli of curves (algebraic) |

14K10 | Algebraic moduli of abelian varieties, classification |

14C05 | Parametrization (Chow and Hilbert schemes) |

20G15 | Linear algebraic groups over arbitrary fields |