Algebraic surfaces and holomorphic vector bundles
This book gives a comprehensive account of the theory of complex projective algebraic surfaces and, simultaneously, of the present state of knowledge concerning the classification theory of holomorphic vector bundles over those surfaces. Basically, it is designed as a textbook for advanced graduate students in those areas of complex geometry that are interspersed with methods of algebraic geometry, for instance such as algebraic geometry itself, gauge theory in global analysis, four-manifold topology, quantum field theory in theoretical physics, and others. On the other hand, with regard to its thoroughness, comprehensiveness and up-to-dateness, this text does provide much more than an excellent introduction to the theory of complex algebraic surfaces and holomorphic vector bundles over them. In fact it also represents a respectable work of reference for researchers and teachers in these areas of mathematics and physics, all the more as there are still only very few reference books on vector bundles available within the existing literature. The book grew out of courses that the author taught, over the past ten years, at several American mathematical institutes. Originally, the goal of these lectures was to acquaint researchers in four-manifold topology with the classification of algebraic surfaces and with methods for describing moduli spaces of holomorphic vector bundles over algebraic surfaces, and that with a special view toward computing Donaldson invariants for four-manifolds. This approach to investigating the topology of smooth four-manifolds is described, in great variety, in the monograph ``Smooth four-manifolds and complex surfaces'' written by \textit{R. Friedman} and \textit{J. W. Morgan} in 1994 (Zbl 0817.14017). Although the focus of four-manifold topology has shifted since that time more and more to the purely analytic-topological aspects connected with the recently discovered Seiberg-Witten invariants, the author's method of teaching the theory of algebraic surfaces and vector bundles remains highly interesting and valuable on its own. Namely, a novel (and unique) feature of the book is provided by the fact that it offers an integrated approach to both algebraic surface theory and the study of vector bundles on curves and surfaces, which certainly makes these two subjects much more conceivable and accessible for the interested reader. While the two subjects remain separate through the first few chapters and are studied in alternate chapters they become increasingly interconnected, even methodically and proof-technically dependend on each other, as the text progresses. For instance, vector bundles on algebraic curves are studied to describe the geometry of ruled surfaces, and then reappear later in the proof of F. Bogomolov's inequality for stable vector bundles on algebraic surfaces. In turn, this inequality is applied to describe canonical embeddings of surfaces via I. Reider's method and E. Bombieri's theorem. Similarly, ruled and elliptic surfaces are discussed in great detail, particularly so by using vector bundles on curves, and then the geometry of vector bundles over such surfaces is thoroughly analyzed. As for the precise contents, the entire text consists of ten chapters. These are arranged as follows: 1. Curves on a surface (invariants of surfaces, curves as divisors on surfaces, adjunction and arithmetic genus, the Riemann-Roch theorem for surfaces, Grothendieck's algebraic proof of the Hodge index theorem, ample and nef divisors); 2. Coherent sheaves (coherent sheaves and Chern classes, rank-2 bundles and sub-line bundles, elementary modifications, singularities of coherent sheaves, torsion-free and reflexive sheaves, double covers, related commutative algebra); 3. Birational geometry (blowing-up, the Castelnuovo criterion and factorization of birational morphisms, minimal models of surfaces, more general contractions of surfaces); 4. Stability of vector bundles over curves and surfaces (Mumford-Takemoto stability, stable bundles on curves and \(\mathbb{P}^2\), Gieseker stability, semi-stable sheaves, the differential geometry of stable vector bundles); 5. Some examples of surfaces (rational ruled surfaces, general ruled surfaces, linear systems of cubics, \(K3\) surfaces); 6. Vector bundles over ruled surfaces (special ample divisors, local and global moduli of vector bundles, a Zariski-open subset in the moduli space of semi-stable vector bundles); 7. An introduction to elliptic surfaces (elliptic fibrations, singular fibers, invariants of elliptic surfaces, elliptic surfaces with sections and Weierstrass models, general elliptic surfaces, the fundamental group); 8. Vector bundles over elliptic surfaces (stable bundles over singular curves, stable bundles over elliptic curves, moduli of stable bundles over elliptic surfaces, an overview of Donaldson invariants of smooth differentiable 4-manifolds, moduli spaces of vector bundles via extensions, vector bundles with trivial determinant, even fiber degree and multiple fibers); 9. Bogomolov's inequality and applications (Bogomolov's inequality for stable rank-bundles over surfaces, the theorems of Bombieri and Reider, symmetric powers of vector bundles on curves, restriction theorems, Galois descent theory); 10. Outline of classification of algebraic surfaces and of stable vector bundles (outline of the birational classification of surfaces, Castelnuovo's theorem, the Albanese map, proofs of the main classification theorems, the Castelnuovo-de Franchis theorem, classification of threefolds, classification of vector bundles over general surfaces). Many of the results on vector bundles over surfaces appear here for the first time in textbook form, and this makes the book particularly useful and accessible for graduate students. For further reading, the reader is referred to the more specialized (and advanced) monographs by \textit{C. Okonek}, \textit{M. Schneider} and \textit{H. Spindler} on ``Vector bundles over complex projective spaces'' Prog. Math. 3 (1980; Zbl 0438.32016), by \textit{C. S. Seshadri} on vector bundles over curves [cf. ``Fibrés vectoriels sur les courbes algébriques, ``Astérisque 96 (1982; Zbl 0517.14008)], and by \textit{D. Huybrechts} and \textit{M. Lehn} on the general theory of moduli spaces of sheaves [cf. ``The geometry of moduli spaces of sheaves'', Aspects Math. E 31 (1997; Zbl 0872.14002)]. Apart from providing an integrated approach to complex algebraic surfaces and the study of vector bundles over them, the book fascinates by its strong emphasis on examples and guided exercises. There are over 100 exercises which form an organic, integral part of the text, and which are extremely helpful for the reader in acquiring the vast knowledge that the book imparts. As for the prerequisites, the reader should at least have had a thorough first-year course in algebraic geometry, perhaps at the level of \textit{R. Hartshorne}'s book ``Algebraic geometry'' [see e.g. the 3rd edition 1983; Zbl 0531.14001)]. Altogether, this textbook is a highly welcome addition to the existing standard texts on complex algebraic surfaces and to the few monographs on vector bundles over algebraic varieties.
moduli spaces of holomorphic vector bundles over algebraic surfaces
scientific article; zbMATH DE number 1134984