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**The geometry of moduli spaces of sheaves.
2nd ed.**
*(English)*
Zbl 1206.14027

Cambridge: Cambridge University Press (ISBN 978-0-521-13420-0/pbk). xviii, 325 p. £ 29.99; $ 50.00; $ 40.00/e-book (2010).

The first edition of this monograph on semistable sheaves and their moduli spaces appeared thirteen years ago, and has been extensively reviewed [Aspects of Mathematics. E 31. Braunschweig: Vieweg. xiv, 269 p. (1997; Zbl 0872.14002)] at that time. Due to its fundamental character, topicality, and expository mastery, the book quickly became both a highly regarded standard text and a basic reference in this very active field of current algebro-geometric research.

Now, as the first edition of this great source book has been out of print for some years, the authors have acceded to the frequent requests for a new edition by presenting the volume under review.

In this second edition of their popular monograph, the authors have left the eleven chapters, together with the respective appendices, of the well-proven text of the original widely intact. However, a number of inaccuracies and mistakes occurring in the first edition have been corrected, and the presentation has been improved at various places, too. But more importantly, the authors have used the opportunity of this second edition of the book to enlarge the text by a substantial amount of additional material.

More precisely, in view of the fact that the subject has undergone various crucial developments during the past fifteen years, the authors have updated the book by means of further appendices. In the present second edition, virtually each chapter contains a new appendix entitled “Further comments”. Here, without striving for completeness nor exhibiting technical details, the authors briefly discuss some new aspects and advances in the theory of semistable sheaves and their moduli spaces, thereby reflecting some of the more recent research topics in this context and providing valuable guidance with regard to the extensive current literature, related to them. The material touched upon in the nine additional appendices refers to recent work on Grothendieck’s Quot-schemes and their generalizations, moduli spaces of semistable sheaves in positive characteristic, moduli spaces of \(G\)-principal bundles over a scheme, moduli spaces of decorated sheaves, moduli spaces of complexes of coherent sheaves (in the context of derived categories), moduli of twisted sheaves, Hilbert schemes of \(K3\) surfaces, the phenomenon of strange duality in the cohomology of moduli spaces of sheaves on algebraic surfaces, the calculation of numerical invariants for moduli spaces of semistable sheaves on various types of algebraic surfaces, and many other related topics. Accordingly, the bibliography has been updated to a large extent. In fact, the additional list of new references contains nearly 200 titles published in the course of the last twelve years, which makes the overall bibliography of the book now comprise the enormous number of 456 references of highly topical character.

Altogether, it must be seen as a more than gratifying circumstance that this unique standard text on the geometry of moduli spaces of sheaves on algebraic varieties has been made available again ,and that even in revised, enlarged and updated form. The appraisal uttered in the review of the first edition [Zbl 0872.14002] of this masterly book could only be repeated, and therefore may it suffice to refer to it in this place. However, both in retrospect and in view of the ameliorations, depicted above, it should be emphasized that this marvellous text is of fundamental importance in the current expository literature on the subject. Serving as a perfect introduction for beginners in the field, an excellent guide to the forefront of research in various directions, a valuable reference for active researchers, and as an abundant source of inspiration for mathematicians and physicists likewise, this book will certainly maintain both its particular significance and its indispensability for further generations of researchers in the field of algebraic sheaves (or vector bundles) and their moduli spaces.

Now, as the first edition of this great source book has been out of print for some years, the authors have acceded to the frequent requests for a new edition by presenting the volume under review.

In this second edition of their popular monograph, the authors have left the eleven chapters, together with the respective appendices, of the well-proven text of the original widely intact. However, a number of inaccuracies and mistakes occurring in the first edition have been corrected, and the presentation has been improved at various places, too. But more importantly, the authors have used the opportunity of this second edition of the book to enlarge the text by a substantial amount of additional material.

More precisely, in view of the fact that the subject has undergone various crucial developments during the past fifteen years, the authors have updated the book by means of further appendices. In the present second edition, virtually each chapter contains a new appendix entitled “Further comments”. Here, without striving for completeness nor exhibiting technical details, the authors briefly discuss some new aspects and advances in the theory of semistable sheaves and their moduli spaces, thereby reflecting some of the more recent research topics in this context and providing valuable guidance with regard to the extensive current literature, related to them. The material touched upon in the nine additional appendices refers to recent work on Grothendieck’s Quot-schemes and their generalizations, moduli spaces of semistable sheaves in positive characteristic, moduli spaces of \(G\)-principal bundles over a scheme, moduli spaces of decorated sheaves, moduli spaces of complexes of coherent sheaves (in the context of derived categories), moduli of twisted sheaves, Hilbert schemes of \(K3\) surfaces, the phenomenon of strange duality in the cohomology of moduli spaces of sheaves on algebraic surfaces, the calculation of numerical invariants for moduli spaces of semistable sheaves on various types of algebraic surfaces, and many other related topics. Accordingly, the bibliography has been updated to a large extent. In fact, the additional list of new references contains nearly 200 titles published in the course of the last twelve years, which makes the overall bibliography of the book now comprise the enormous number of 456 references of highly topical character.

Altogether, it must be seen as a more than gratifying circumstance that this unique standard text on the geometry of moduli spaces of sheaves on algebraic varieties has been made available again ,and that even in revised, enlarged and updated form. The appraisal uttered in the review of the first edition [Zbl 0872.14002] of this masterly book could only be repeated, and therefore may it suffice to refer to it in this place. However, both in retrospect and in view of the ameliorations, depicted above, it should be emphasized that this marvellous text is of fundamental importance in the current expository literature on the subject. Serving as a perfect introduction for beginners in the field, an excellent guide to the forefront of research in various directions, a valuable reference for active researchers, and as an abundant source of inspiration for mathematicians and physicists likewise, this book will certainly maintain both its particular significance and its indispensability for further generations of researchers in the field of algebraic sheaves (or vector bundles) and their moduli spaces.

Reviewer: Werner Kleinert (Berlin)

### MSC:

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

14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |

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

14J60 | Vector bundles on surfaces and higher-dimensional varieties, and their moduli |