##
**Smooth four-manifolds and complex surfaces.**
*(English)*
Zbl 0817.14017

Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 27. Berlin: Springer-Verlag. x, 520 p. (1994).

In the early 1960’s there were significant breakthroughs in two branches in mathematics. On the one hand, after the rigorous foundation of algebraic geometry and complex-analytic geometry by sheaf-theoretic and cohomological methods in the foregoing decade, K. Kodaira succeeded in giving the classical theory of complex algebraic surfaces the necessary precision and solid base. He not only re-established and extended the Enriques’ classification of surfaces, including also compact non- algebraic surfaces, but he also created (together with D. C. Spencer) the deformation-theoretic framework for the moduli theory of special classes of complex surfaces. In particular, he did the first steps towards the classification of K3-surfaces and began the systematic study of (non-) algebraic elliptic surfaces, a topic that had not been taken up by the great old Italian geometers.

On the other hand, and almost simultaneously, S. Smale made a great progress towards the old and difficult problem of classifying all possible smooth and/or topological structures on a given differentiable manifold. He established the generalized Poincaré conjecture for manifolds of dimension at least five and, in the sequel, proved the famous \(h\)-cobordism theorem. Based on these results, the main classification scheme for manifolds of dimension \(\geq 5\) could be developed, in the following decade, and by the mid 1970’s it had reached its nearly complete shape. The major effort then shifted to the remaining cases of three-manifolds and four-manifolds. Due to the work of W. Thurston in the early 1980’s, there is now a possible classification scheme in dimension three, too, at least in a concrete conjectural form, which may serve as the guiding idea for completing the classification theory of three-manifolds. The case of four-manifolds, however, still remains exceptional and fairly mysterious, even with regard to the existence of guiding conjectural outlines as in the three-dimensional situation. – One possible approach to a better understanding of four- manifolds is provided by the fact that some of them occur as the topological objects underlying smooth complex surfaces, for which the well-developed Kodaira classification theory is available. On the other hand, many still unsolved problems on the topology and differential geometry of algebraic surfaces lead to the methods of real four-manifold theory.

This link between smooth complex surfaces and real four-manifolds is the topic of the present book. The basic method used by the authors is S. Donaldson’s recent approach of studying four-manifolds by introducing a suitable gauge theory for them and, along this way, defining new smooth invariants for four-manifolds, which possibly are appropriate substitutes for the usual homotopy or homeomorphism invariants that definitely fail in the exceptional case of dimension four. Donaldson’s so-called “polynomial invariants” are not easily accessible to computation, and in general very little is known about their structure and classifying significance. On the other hand, Donaldson’s gauge theory methods are particularly well-suited to study four-manifolds underlying complex algebraic surfaces. Namely, according to a theorem S. K. Donaldson [Proc. Lond. Math. Soc., III. Ser. 50, 1-26 (1985; Zbl 0529.53018)], for an algebraic surface the (gauge-theoretic) moduli space of anti-self-dual connections is equivalent to the moduli space of certain stable holomorphic vector bundles on the surface, and that makes it possible to study Donaldson’s polynomial invariants of complex algebraic surfaces by algebro-geometric techniques.

According to the aim of this book, the authors have devoted much of the text to study Donaldson’s invariants for special types of algebraic surfaces and, as a consequence from their results, to discuss new informations about the differential geometry of those kinds of surfaces. Altogether, the central part of this work represents a far-reaching continuation of the (generally very intensive) study of the Donaldson invariants of algebraic surfaces, based upon a subtle combination of methods from the algebro-geometric classification theory and the computation of polynomial invariants. The main result, derived in the course of the text, shows that the smooth classification of algebraic surfaces, unlike their topological classification or the case of manifolds of dimension different from four, is deeply related to their algebro-geometric structure. This unique phenomenon has been proved by the authors in a previous original article [cf. R. Friedman and J. W. Morgan, J. Differ. Geom. 27, No. 2, 297-369 and No. 3, 371- 398 (1988; Zbl 0669.57016 and Zbl 0669.57017)], independently also by C. Okonek and A. Van de Ven [Invent. Math. 86, 357-370 (1986; Zbl 0613.14018)], and these results comprise most of the last two chapters of this book, together with furthergoing results on the topological characterization of algebraic K3-surfaces and (blow-ups of) nonrational elliptic surfaces.

In this regard the present book represents both a research monograph on algebro-geometric methods in the theory of smooth four manifolds, on the one hand, and a textbook introducing the reader into the classification theory of algebraic surfaces and into Donaldson’s gauge theory of smooth four-manifolds, on the other hand. The text consists of seven chapters which are arranged as follows.

Chapter 1 resembles, in a fairly self-contained manner and for the convenience of the reader, the Kodaira classification of algebraic surfaces. The authors have put special emphasis on elliptic surfaces, since those play a crucial role in the sequel, and because the literature on elliptic surfaces is much too vast as for reasonable references. This chapter also contains some new results on the deformation behavior of elliptic surfaces. – Chapter 2 discusses the topology of elliptic surfaces and provides, in this way, the perspective of the later differentiable classification of algebraic surfaces. – Chapter 3 deals with the basics of Donaldson’s gauge theory for four-manifolds and, in particular, with the definition of his polynomial invariants. This includes material on the space of connections, moduli of anti-self-dual connections, and various kinds of Donaldson polynomial invariants. – Chapter 4 is devoted to the moduli spaces of stable holomorphic vector bundles over complex algebraic surfaces and the gauge theory corresponding to the Kähler differential geometry of surfaces. – Then, in chapter 5, the authors develop explicit methods for computing Donaldson polynomials of algebraic surfaces, in terms of algebraic cohomology classes. In an appendix, this method is compared to different technical approaches based on various notions of determinants of line bundles (due to Atiyah-Singer, Bismut-Gillet-Soulé, and others). – Chapter 6 switches back to four-manifold theory. The objects under consideration are particular four-manifolds, namely those with so-called big diffeomorphism groups with respect to a given cohomology class, and the structure of their Donaldson polynomial invariants. – The concluding chapter 7 provides explicit computations of Donaldson polynomials for elliptic surfaces. Using the techniques developed in chapters 5 and 6, together with some standard algebraic geometry, the authors derive their main results (explained above) and some conjectural hints for further research in this area.

The authors have done their very best to present a comprehensive, detailed and self-contained combination of a research monograph and a textbook on this fascinating, utmost actual topic in mathematics today. Requiring only the basics of algebraic geometry and gauge theory in differential geometry, the content of the book leads the reader to the forefront of contemporary research in both complex surface theory and the topology of smooth four-manifolds. Already the introduction is written in a very careful, motivating and enlightening way, and each section comes with its own detailed introduction. The style of writing is masterly lucid, rigorous and comprehensible. As for more background material, the reader should be referred to the classics “Compact complex surfaces” by W. Barth, C. Peters and A. Van de Ven [Ergeb. Math. Grenzgeb., 3. Folge, Bd. 4 (1984; Zbl 0718.14023)] and “The geometry of four-manifolds” by S. K. Donaldson and P. B. Kronheimer (1990), but the essential basic material is actually contained in the beautiful book under review.

On the other hand, and almost simultaneously, S. Smale made a great progress towards the old and difficult problem of classifying all possible smooth and/or topological structures on a given differentiable manifold. He established the generalized Poincaré conjecture for manifolds of dimension at least five and, in the sequel, proved the famous \(h\)-cobordism theorem. Based on these results, the main classification scheme for manifolds of dimension \(\geq 5\) could be developed, in the following decade, and by the mid 1970’s it had reached its nearly complete shape. The major effort then shifted to the remaining cases of three-manifolds and four-manifolds. Due to the work of W. Thurston in the early 1980’s, there is now a possible classification scheme in dimension three, too, at least in a concrete conjectural form, which may serve as the guiding idea for completing the classification theory of three-manifolds. The case of four-manifolds, however, still remains exceptional and fairly mysterious, even with regard to the existence of guiding conjectural outlines as in the three-dimensional situation. – One possible approach to a better understanding of four- manifolds is provided by the fact that some of them occur as the topological objects underlying smooth complex surfaces, for which the well-developed Kodaira classification theory is available. On the other hand, many still unsolved problems on the topology and differential geometry of algebraic surfaces lead to the methods of real four-manifold theory.

This link between smooth complex surfaces and real four-manifolds is the topic of the present book. The basic method used by the authors is S. Donaldson’s recent approach of studying four-manifolds by introducing a suitable gauge theory for them and, along this way, defining new smooth invariants for four-manifolds, which possibly are appropriate substitutes for the usual homotopy or homeomorphism invariants that definitely fail in the exceptional case of dimension four. Donaldson’s so-called “polynomial invariants” are not easily accessible to computation, and in general very little is known about their structure and classifying significance. On the other hand, Donaldson’s gauge theory methods are particularly well-suited to study four-manifolds underlying complex algebraic surfaces. Namely, according to a theorem S. K. Donaldson [Proc. Lond. Math. Soc., III. Ser. 50, 1-26 (1985; Zbl 0529.53018)], for an algebraic surface the (gauge-theoretic) moduli space of anti-self-dual connections is equivalent to the moduli space of certain stable holomorphic vector bundles on the surface, and that makes it possible to study Donaldson’s polynomial invariants of complex algebraic surfaces by algebro-geometric techniques.

According to the aim of this book, the authors have devoted much of the text to study Donaldson’s invariants for special types of algebraic surfaces and, as a consequence from their results, to discuss new informations about the differential geometry of those kinds of surfaces. Altogether, the central part of this work represents a far-reaching continuation of the (generally very intensive) study of the Donaldson invariants of algebraic surfaces, based upon a subtle combination of methods from the algebro-geometric classification theory and the computation of polynomial invariants. The main result, derived in the course of the text, shows that the smooth classification of algebraic surfaces, unlike their topological classification or the case of manifolds of dimension different from four, is deeply related to their algebro-geometric structure. This unique phenomenon has been proved by the authors in a previous original article [cf. R. Friedman and J. W. Morgan, J. Differ. Geom. 27, No. 2, 297-369 and No. 3, 371- 398 (1988; Zbl 0669.57016 and Zbl 0669.57017)], independently also by C. Okonek and A. Van de Ven [Invent. Math. 86, 357-370 (1986; Zbl 0613.14018)], and these results comprise most of the last two chapters of this book, together with furthergoing results on the topological characterization of algebraic K3-surfaces and (blow-ups of) nonrational elliptic surfaces.

In this regard the present book represents both a research monograph on algebro-geometric methods in the theory of smooth four manifolds, on the one hand, and a textbook introducing the reader into the classification theory of algebraic surfaces and into Donaldson’s gauge theory of smooth four-manifolds, on the other hand. The text consists of seven chapters which are arranged as follows.

Chapter 1 resembles, in a fairly self-contained manner and for the convenience of the reader, the Kodaira classification of algebraic surfaces. The authors have put special emphasis on elliptic surfaces, since those play a crucial role in the sequel, and because the literature on elliptic surfaces is much too vast as for reasonable references. This chapter also contains some new results on the deformation behavior of elliptic surfaces. – Chapter 2 discusses the topology of elliptic surfaces and provides, in this way, the perspective of the later differentiable classification of algebraic surfaces. – Chapter 3 deals with the basics of Donaldson’s gauge theory for four-manifolds and, in particular, with the definition of his polynomial invariants. This includes material on the space of connections, moduli of anti-self-dual connections, and various kinds of Donaldson polynomial invariants. – Chapter 4 is devoted to the moduli spaces of stable holomorphic vector bundles over complex algebraic surfaces and the gauge theory corresponding to the Kähler differential geometry of surfaces. – Then, in chapter 5, the authors develop explicit methods for computing Donaldson polynomials of algebraic surfaces, in terms of algebraic cohomology classes. In an appendix, this method is compared to different technical approaches based on various notions of determinants of line bundles (due to Atiyah-Singer, Bismut-Gillet-Soulé, and others). – Chapter 6 switches back to four-manifold theory. The objects under consideration are particular four-manifolds, namely those with so-called big diffeomorphism groups with respect to a given cohomology class, and the structure of their Donaldson polynomial invariants. – The concluding chapter 7 provides explicit computations of Donaldson polynomials for elliptic surfaces. Using the techniques developed in chapters 5 and 6, together with some standard algebraic geometry, the authors derive their main results (explained above) and some conjectural hints for further research in this area.

The authors have done their very best to present a comprehensive, detailed and self-contained combination of a research monograph and a textbook on this fascinating, utmost actual topic in mathematics today. Requiring only the basics of algebraic geometry and gauge theory in differential geometry, the content of the book leads the reader to the forefront of contemporary research in both complex surface theory and the topology of smooth four-manifolds. Already the introduction is written in a very careful, motivating and enlightening way, and each section comes with its own detailed introduction. The style of writing is masterly lucid, rigorous and comprehensible. As for more background material, the reader should be referred to the classics “Compact complex surfaces” by W. Barth, C. Peters and A. Van de Ven [Ergeb. Math. Grenzgeb., 3. Folge, Bd. 4 (1984; Zbl 0718.14023)] and “The geometry of four-manifolds” by S. K. Donaldson and P. B. Kronheimer (1990), but the essential basic material is actually contained in the beautiful book under review.

Reviewer: W.Kleinert (Berlin)

### MSC:

14J15 | Moduli, classification: analytic theory; relations with modular forms |

14J27 | Elliptic surfaces, elliptic or Calabi-Yau fibrations |

32G13 | Complex-analytic moduli problems |

57R55 | Differentiable structures in differential topology |

58D27 | Moduli problems for differential geometric structures |

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