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Advances in moduli theory. Transl. from the Japanese by Yuji Shimizu and Kenji Ueno. (English) Zbl 0987.14001
Translations of Mathematical Monographs. 206. Providence, RI: American Mathematical Society (AMS). xix, 300 p. (2002).
The present book is the English translation of the Japanese edition “Moduli theory. 3” [Iwanami Shoten Publ., Tokyo 1999]. The foregoing volumes, “Moduli Theory 1 and 2” Cambridge Tracts in Mathematics. 148, Cambridge University Press. (2001; Zbl 1033.14008), are written by S. Mukai, and their English translation is going to be publishing by Cambridge University Press, UK, in the near future.
Whereas the first two parts by S. Mukai focus on the algebraic aspects of moduli theory, the volume under review treats various topics of this subject from the viewpoint of complex analytic geometry. In this vein, the present book is fairly independent of S. Mukai’s texts and can be studied separately.
Basically, the author’s main goal was to provide a concise introduction to modern complex analytic classification theory, together with some of its very recent, spectacular applications to conformal quantum field theory in mathematical physics. The text is aimed at graduate and upper-level undergraduate students in algebraic geometry and complex analysis, on the one hand, yet just as well suited for physicists using algebro-geometric and complex analytic methods in the study of modern quantum field theories, on the other hand.
As for the selected material and its presentation, the authors have followed the line of historical development in complex analytic deformation and moduli theory, beginning from its modern foundation via sheaf-theoretic and cohomological concepts in the 1950’s and 1960’s and leading to the more recent, powerful framework of (mixed) Hodge structures and their variations.
This purely mathematical part of the text, which constitutes about two thirds of the entire book, is followed by a rather comprehensive chapter entitled “Conformal field theory”. In this concluding chapter, which forms the remaining third of the book, the authors describe some recent advances of moduli theory in the guise of applications to non-abelian conformal quantum field theory in mathematical physics.
More precisely, the book consists of four chapters, the concrete contents of which are as follows.
Chapter 1 comes with the title “Kodaira-Spencer mapping” and discusses the deformation theory of complex structures initiated by K. Kodaira and D. C. Spencer in the late 1950’s. This material also includes, apart from the local aspects, a thorough explanation of the Kuranishi family as well as a brief analysis of the infinitesimal deformation of holomorphic principal bundles of a complex Lie group over a given complex manifold.
Chapter 2, entitled “Torelli’s theorem”, turns to the classification theory of compact Riemann surfaces of genus $$g$$ (or to smooth projective algebraic curves of genus $$g$$, respectively), i.e., to the very classical part of analytic moduli theory. This chapter gives the beautiful, evergreen standard introduction to complex tori, complex abelian varieties, theta functions, Jacobians of compact Riemann surfaces, Riemann matrices, period mappings, moduli spaces for compact Riemann surfaces, and Torelli’s theorem. As for the various great proofs of Torelli’s theorem, the authors provide a sketch of A. Andreotti’s fascinating geometrical proof and, in the sequel, present G. Martens’s amazingly elementary proof in full detail.
Chapter 3, entitled “Period mappings and Hodge theory”, is devoted to the generalization of the classical theory of abelian integrals for complex algebraic curves to higher-dimensional complex manifolds. More precisely, this chapter describes the Hodge decomposition for a Kähler manifold, its axiomatic generalization leading to the concept of a (polarized) Hodge structure, Griffiths’s theory of variations of Hodge structures in families of complex manifolds, the allied notion of the Gauss-Manin connection, and Griffiths’s construction of the classifying space for Hodge structures, the so-called period domain. This is then followed by a discussion of the period mapping for a smooth family of projective varieties, together with a (basically example-driven) survey on the various Torelli-type problems related to the period mapping. The concluding part of this chapter deals with Deligne’s concept of a mixed Hodge structure, some of the main results related to this conceptual framework (such as the monodromy theorem, the nilpotent orbit theorem by W. Schmid, the existence of limit Hodge structures), and – at the end of this chapter – with the degeneration of mixed Hodge structures. Due to the vast theory of Hodge structures developed over the past four decades, the authors have mainly focused on explaining and illustrating the most fundamental concepts and theorems, instead of giving complete proofs and all the technicalities in full detail. In this regard, chapter 3 appears somewhat more sketchy than the foregoing two chapters, but this is didactically absolutely justified and shows, like the entire text, the authors’ masterly skill in writing an introduction to such a highly advanced topic.
Chapter 4 has been given the title “Conformal field theory” and turns to the applications of complex analytic moduli theory to modern mathematical physics. More precisely, as an application of the geometry of moduli spaces for curves, the authors discuss a model for a non-abelian conformal field theory. This model was elaborated by A. Tsuchiya, K. Ueno and Y. Yamada in 1989 [Adv. Stud. Pure Math. 19, 459-566 (1989; Zbl 0696.17010)] and is presented here in full detail, together with some very recent advances in conformal quantum field theory and algebraic geometry, respectively.
After formulating the requirements for a so-called “holomorphic conformal field theory” in terms of $$N$$-pointed compact Riemann surfaces, the authors describe stable pointed curves and their moduli spaces, then conformal blocks and their sheaf-theoretic interpretation, the infinitesimal geometry of the Knizhnik-Zamolodchikov equations for this holomorphic conformal field theory, and conclude the text with a discourse on the famous, only recently established Verlinde formula. This includes, of course, an explanation of the relevant facts from the theory of moduli spaces of vector bundles over a complex algebraic curve.
In an appendix added to the text, the authors point out various historical aspects, further developments, prospects, and remaining problems in both analytic moduli theory and quantum field theory. These remarks are enhanced, like the entire text itself, by ample references to the current literature for complementary and further reading.
A particular feature of this utmost useful introductory textbook on analytic moduli problems and their role in mathematical physics is the vast amount of instructive examples and guiding remarks pervading the entire text. For the convenience of the reader, the authors have added a summary of the main concepts and results to each single chapter. As for the problems and exercises scattered in the text, complete solutions for them are compiled at the end of the book, likewise for the convenience of the (unexperienced) reader.
All together, this book provides a masterly introduction to the analytical aspects of moduli theory, in its modern and advanced setting, and just as well to its recent, spectacular applications in quantum physics. The text is perfectly suited for a first profound study of these subjects, though on an already rather advanced level, and, in that manner, it is certainly an excellent source for courses, seminars, and individual studies. Also, the book appears as an overpowering invitation to these fascinating and central topics of contemporary mathematics and physics. In combination with S. Mukai’s “Moduli theory 1 and 2”, the translation of which will appear very soon, the book under review must be seen as a highly welcome enhancement of the yet very few standard texts on moduli theory in general.

##### MSC:
 14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry 14D20 Algebraic moduli problems, moduli of vector bundles 81-02 Research exposition (monographs, survey articles) pertaining to quantum theory 81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics 14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) 32G20 Period matrices, variation of Hodge structure; degenerations 32G13 Complex-analytic moduli problems