The geometry of moduli spaces of sheaves.

*(English)*Zbl 0872.14002
Aspects of Mathematics. E 31. Braunschweig: Vieweg. xiv, 269 p. (1997).

The present book grew out of a series of lectures delivered by the two authors at the Summer School 1995 of the Graduiertenkolleg “Geometry and nonlinear analysis” at Humboldt University in Berlin. While the original lectures were designed to discuss some of the recent results on the geometry of moduli spaces of (semi-)stable coherent sheaves on an algebraic surface, the text at hand is a considerably elaborated and extended version of the initial notes. The outcome of the authors’ rewarding and admirable effort at completing their lecture notes is now a book that serves several purposes at the same time. On the one hand, and in regard of its first part, it provides a textbook-like introduction to the theory of (semi-)stable coherent sheaves over arbitrary algebraic varieties and to their moduli spaces. On the other hand, mainly in view of its second part, the text has the character of both a research monograph and a comprehensive survey on some very recent results on those moduli spaces of (semi-)stable sheaves over (special) algebraic surfaces.

In both aspects, this book is rather unique in the existing literature on the classification theory of sheaves and vector bundles. Namely, for the first time in this central current area of research in algebraic geometry, a successful attempt has been undertaken to develop both the general, conceptual and methodical framework and the present state of knowledge in one of the most important special cases in a systematic, detailed, nearly complete and didactically processed presentation.

The text is divided into two major parts. After a careful introduction, which provides several motivations for studying sheaves on algebraic surfaces, in particular with a view to their significance in the differential geometry of four-dimensional manifolds and in gauge field theory (e.g., via Donaldson polynomials and Seiberg-Witten invariants), part I is devoted to the general theory of semi-stable sheaves and their moduli spaces. Chapter 1 introduces the basic concept of semi-stability for coherent sheaves over algebraic varieties, in the sense of D. Gieseker as well as in the (original) version of Mumford-Takemoto, and the fundamental material on Harder-Narasimhan filtrations, Jordan-Hölder filtrations, \(S\)-equivalence for semi-stable sheaves, and boundedness conditions. Flat families of sheaves, Grothendieck’s Quot-scheme, the deformation theory of flags of sheaves, and Maruyama’s openness-of-stability theorem are discussed in chapter 2, while chapter 3 deals with the most general form of the so-called Grauert-Mülich theorem and its application in establishing the boundedness of the set of semi-stable sheaves.

Moduli spaces for semi-stable sheaves, in their local and global aspects, is the subject of chapter 4. The authors discuss in detail C. Simpson’s more recent approach to the construction of these moduli spaces, together with the related general facts from geometric invariant theory, and sketch the original construction by D. Gieseker and M. Maruyama likewise in an appendix. Furthermore, deformation theory is used to analyze the local structure of these moduli spaces, including dimension bounds and estimates for the expected dimension in the case of an algebraic surface. In another appendix the authors give an outlook to their own research contributions, in that they briefly describe moduli for “decorated sheaves” [cf. D. Huybrechts and M. Lehn, Int. J. Math. No. 2, 297-324 (1995; Zbl 0865.14004)]. This topic, though not systematically treated in the text, has recently found spectacular applications in conformal quantum field theory (e.g., in M. Thaddeus’s proof of the famous Verlinde formula) and in non-abelian Seiberg-Witten theory.

The second part of the book, starting with chapter 5, mainly focuses on moduli spaces of semi-stable sheaves on algebraic surfaces. At first, the authors present various construction methods for stable vector bundle on surfaces, including Serre’s correspondence between rank-2 vector bundles and codimension-2 subschemes, Maruyama’s method of elementary transformations, and some illustrating examples. The geometry of moduli spaces of semi-stable sheaves on K3 surfaces is thoroughly explained in chapter 6, where in particular some very recent results by S. Mukai, A. Beauville, L. Göttsche-D. Huybrechts, K. O’Grady, J. Li, G. Ellingsrud-M. Lehn, and others are systematically compiled. Chapter 7 deals with the restriction of sheaves on surfaces to curves, focusing on the related work of H. Flenner, F. Bogomolov, and V. Mehta-A. Ramanathan in the 1980’s. In chapter 8, the authors turn the attention to line bundles on moduli spaces and their Picard groups. The construction of determinantal line bundles and ampleness results for special line bundles on moduli spaces are presented by essentially following the approaches of J. Le Potier (1989) and J. Li (1993). As an application, the authors provide a profound comparison between the (algebraic) Gieseker-Maruyama moduli spaces of semi-stable vector bundles and the (analytic) Donaldson-Uhlenbeck compactification of the moduli spaces of Mumford-stable bundles. Chapter 9 is almost entirely devoted to K. O’Grady’s recent work on the irreducibility and generic smoothness of moduli spaces for vector bundles on projective surfaces [cf. K. O’Grady, Invent. Math. 123, No, 1, 141-207 (1996; Zbl 0869.14005)] and the related results by D. Gieseker and J. Lie [J. Am. Math. Soc. 9, 107-151 (1996; Zbl 0864.14005)].

Chapter 10, entitled “Symplectic structures”, turns to differential forms on moduli spaces of stable sheaves on surfaces. After a lucid survey of the technical background material such as Atiyah classes, trace maps, cup products, the Kodaira-Spencer map, etc., the authors describe the tangent bundle of the smooth part of a moduli space by means of the universal family of vector bundles. Then, via the explicit construction of closed differential forms on moduli spaces, Mukai’s theorem on the existence of a non-degenerate symplectic structure on the moduli space of stable sheaves on a K3 surface is derived.

The concluding chapter 11 deals with the birational properties of moduli spaces of semi-stable sheaves on surfaces. The main result presented here is a simplified proof of J. Li’s recent theorem [Invent. Math. 115, No. 1, 1-40 (1994; Zbl 0799.14015)] stating that moduli spaces of semi-stable sheaves on surfaces of general type are also of general type. Other results on the birational type of such moduli spaces are surveyed in a brief sub-section, and the treatise concludes with two instructive examples showing how the Serre correspondence can be used to obtain information about the birational structure of moduli spaces of sheaves on a K3 surface. Actually, both examples are variations on two recent theorems due to T. Nakashima (1993) and K. O’Grady (1995), respectively, and their discussion is based upon an elegant combination of the results from chapter 8 and 10 in the book.

Altogether, the present text fascinates by comprehensiveness, rigor, profundity, up-to-dateness and methodical mastery. The bibliography comprises 263 references, most of which are really referred to in the course of the text. Each chapter comes with its own specific introduction and, always at the end, with a list of extra comments, hints to the original literature, and remarks on related topics, further developments and current research problems. The authors have successfully tried to keep the presentation of this highly advanced material as self-contained as possible, so that the text should be accessible for readers with a solid background in algebraic geometry. Active researchers in the field will appreciate this book as a valuable source and reference for their work.

In both aspects, this book is rather unique in the existing literature on the classification theory of sheaves and vector bundles. Namely, for the first time in this central current area of research in algebraic geometry, a successful attempt has been undertaken to develop both the general, conceptual and methodical framework and the present state of knowledge in one of the most important special cases in a systematic, detailed, nearly complete and didactically processed presentation.

The text is divided into two major parts. After a careful introduction, which provides several motivations for studying sheaves on algebraic surfaces, in particular with a view to their significance in the differential geometry of four-dimensional manifolds and in gauge field theory (e.g., via Donaldson polynomials and Seiberg-Witten invariants), part I is devoted to the general theory of semi-stable sheaves and their moduli spaces. Chapter 1 introduces the basic concept of semi-stability for coherent sheaves over algebraic varieties, in the sense of D. Gieseker as well as in the (original) version of Mumford-Takemoto, and the fundamental material on Harder-Narasimhan filtrations, Jordan-Hölder filtrations, \(S\)-equivalence for semi-stable sheaves, and boundedness conditions. Flat families of sheaves, Grothendieck’s Quot-scheme, the deformation theory of flags of sheaves, and Maruyama’s openness-of-stability theorem are discussed in chapter 2, while chapter 3 deals with the most general form of the so-called Grauert-Mülich theorem and its application in establishing the boundedness of the set of semi-stable sheaves.

Moduli spaces for semi-stable sheaves, in their local and global aspects, is the subject of chapter 4. The authors discuss in detail C. Simpson’s more recent approach to the construction of these moduli spaces, together with the related general facts from geometric invariant theory, and sketch the original construction by D. Gieseker and M. Maruyama likewise in an appendix. Furthermore, deformation theory is used to analyze the local structure of these moduli spaces, including dimension bounds and estimates for the expected dimension in the case of an algebraic surface. In another appendix the authors give an outlook to their own research contributions, in that they briefly describe moduli for “decorated sheaves” [cf. D. Huybrechts and M. Lehn, Int. J. Math. No. 2, 297-324 (1995; Zbl 0865.14004)]. This topic, though not systematically treated in the text, has recently found spectacular applications in conformal quantum field theory (e.g., in M. Thaddeus’s proof of the famous Verlinde formula) and in non-abelian Seiberg-Witten theory.

The second part of the book, starting with chapter 5, mainly focuses on moduli spaces of semi-stable sheaves on algebraic surfaces. At first, the authors present various construction methods for stable vector bundle on surfaces, including Serre’s correspondence between rank-2 vector bundles and codimension-2 subschemes, Maruyama’s method of elementary transformations, and some illustrating examples. The geometry of moduli spaces of semi-stable sheaves on K3 surfaces is thoroughly explained in chapter 6, where in particular some very recent results by S. Mukai, A. Beauville, L. Göttsche-D. Huybrechts, K. O’Grady, J. Li, G. Ellingsrud-M. Lehn, and others are systematically compiled. Chapter 7 deals with the restriction of sheaves on surfaces to curves, focusing on the related work of H. Flenner, F. Bogomolov, and V. Mehta-A. Ramanathan in the 1980’s. In chapter 8, the authors turn the attention to line bundles on moduli spaces and their Picard groups. The construction of determinantal line bundles and ampleness results for special line bundles on moduli spaces are presented by essentially following the approaches of J. Le Potier (1989) and J. Li (1993). As an application, the authors provide a profound comparison between the (algebraic) Gieseker-Maruyama moduli spaces of semi-stable vector bundles and the (analytic) Donaldson-Uhlenbeck compactification of the moduli spaces of Mumford-stable bundles. Chapter 9 is almost entirely devoted to K. O’Grady’s recent work on the irreducibility and generic smoothness of moduli spaces for vector bundles on projective surfaces [cf. K. O’Grady, Invent. Math. 123, No, 1, 141-207 (1996; Zbl 0869.14005)] and the related results by D. Gieseker and J. Lie [J. Am. Math. Soc. 9, 107-151 (1996; Zbl 0864.14005)].

Chapter 10, entitled “Symplectic structures”, turns to differential forms on moduli spaces of stable sheaves on surfaces. After a lucid survey of the technical background material such as Atiyah classes, trace maps, cup products, the Kodaira-Spencer map, etc., the authors describe the tangent bundle of the smooth part of a moduli space by means of the universal family of vector bundles. Then, via the explicit construction of closed differential forms on moduli spaces, Mukai’s theorem on the existence of a non-degenerate symplectic structure on the moduli space of stable sheaves on a K3 surface is derived.

The concluding chapter 11 deals with the birational properties of moduli spaces of semi-stable sheaves on surfaces. The main result presented here is a simplified proof of J. Li’s recent theorem [Invent. Math. 115, No. 1, 1-40 (1994; Zbl 0799.14015)] stating that moduli spaces of semi-stable sheaves on surfaces of general type are also of general type. Other results on the birational type of such moduli spaces are surveyed in a brief sub-section, and the treatise concludes with two instructive examples showing how the Serre correspondence can be used to obtain information about the birational structure of moduli spaces of sheaves on a K3 surface. Actually, both examples are variations on two recent theorems due to T. Nakashima (1993) and K. O’Grady (1995), respectively, and their discussion is based upon an elegant combination of the results from chapter 8 and 10 in the book.

Altogether, the present text fascinates by comprehensiveness, rigor, profundity, up-to-dateness and methodical mastery. The bibliography comprises 263 references, most of which are really referred to in the course of the text. Each chapter comes with its own specific introduction and, always at the end, with a list of extra comments, hints to the original literature, and remarks on related topics, further developments and current research problems. The authors have successfully tried to keep the presentation of this highly advanced material as self-contained as possible, so that the text should be accessible for readers with a solid background in algebraic geometry. Active researchers in the field will appreciate this book as a valuable source and reference for their work.

Reviewer: W.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 |