An introduction to invariants and moduli. Transl. from the Japanese by W. M. Oxbury (Q2898733)

This book is the unaltered reprint of the original hardback edition of Shigeru Mukai's text ``An Introduction to Invariants and Moduli'' [Cambridge Tracts in Mathematics 81. Cambridge: Cambridge University Press (2003; Zbl 1033.14008)], which was published almost ten years ago as the English translation of the author's two-volume set ``Theory of Moduli. I. II'' (Japanese) from 1998/2000.NEWLINENEWLINE After a first reprint of the English hardback edition in 2006, the present second new edition has been made available in paperback form, with a resulting drop in price of nearly forty percent.NEWLINENEWLINE As the well-tried text has been left entirely unchanged, we may utterly refer to our comprehensive review (Zbl 1033.14008) of the original edition, in which both the precise contents and the particular features of this outstanding treatise were depicted in great detail. However, it should be emphasized again that the (still unique) aim of this text is to provide a concise introduction to the geometry of algebraic varieties, on the one hand, and to the principles of algebraic moduli theory on the other, with the focus on moduli spaces and their allied aspects of both classical and geometric invariant theory. In order to keep the whole presentation as introductory and self-contained as possible, the basics of classical algebraic geometry are developed along the way, and as far as necessary for the study of some concrete moduli spaces in algebraic geometry, while a very high value is set on explaining (and illustrating) the conceptual framework of both classical and geometric invariant theory in great profundity, vividness, and topical versality.NEWLINENEWLINE No doubt, S. Mukai's brilliant and masterly introduction to basic algebraic geometry, invariant theory, and algebraic moduli spaces has maintained its unique role in the relevant textbook literature, also after ten years since its first appearance in English, and it certainly must be seen as one of the most important primers in this field. The reviewer can only repeat what he already stated ten years ago, namely that this marvelous text fascinates by its great originality, glaring expertise of the author's, admirable art of easiness and lucidity, topicality, reader-friendliness, and power of inspiration likewise. In this regard, the book remains an excellent introduction to invariant theory and moduli aspects of algebraic geometry, even as a reprint without any up-datings of the bibliography, and an eager reader will still find his way to the forefront of current research in this contemporary, central area of algebraic geometry through this invaluable guide to it. Further generations of students and instructors in algebraic geometry will profit a great deal from Professor Mukai's unique classic, which fortunately has become available again, due to its present faithfully reprinted first paperback edition.