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Lectures on algebraic geometry II. Basic concepts, coherent cohomology, curves and their Jacobians. (English) Zbl 1227.14001

Aspects of Mathematics E 39. Wiesbaden: Vieweg+Teubner (ISBN 978-3-8348-0432-7/hbk). xiii, 365 p. (2011).
The first volume of G. Harder’s “Lectures on algebraic geometry I. Sheaves, cohomology of sheaves, and applications to Riemann surfaces” appeared three years ago [Aspects of Mathematics E 35. Wiesbaden: Vieweg (2008; Zbl 1129.14001)] and was predominantly devoted to those fundamental concepts, methods, and techniques which are absolutely indispensable for formulating and treating algebraic geometry in its scheme-theoretic and cohomological language à la A. Grothendieck and his successors. Actually, two thirds of this first volume dealt with the basic prerequisites from general category theory, abstract homological algebra, sheaf theory, and cohomology theory of sheaves, nevertheless so with a steady prevailing view toward their later applications to algebraic geometry, algebraic topology, and complex-analytic geometry.
The book under review is the long-announced second volume within the author’s respective publication program, which still is intended to culminate in a future third volume devoted to the current active research topic of the cohomology of arithmetic groups.
While the first volume presented the first five chapters of the whole treatise, this second volume completes the introduction to modern algebraic geometry by the subsequent five chapters. Each chapter is divided into several sections and subsections, which certainly enhances the overall lucidity of the entire exposition.
As for the more precise contents, Chapter 6, the first chapter of the current second volume, is titled “Basic Concepts of the Theory of Schemes”. The author develops here the language of schemes in its general and abstract setting. This includes affine schemes and general schemes, quasi-coherent sheaves of modules, algebraic vector bundles on schemes, affine and flat morphisms, relative schemes and base change, the various concepts of points in a scheme, representable functors in algebraic geometry, and an introduction to the theory of descend for later use.
Chapter 7 provides some basic commutative algebra and its use in algebraic geometry. Among the selected topics discussed here are those which are especially related to geometry and algebraic number theory: finitely generated algebras, low-dimensional rings and basic results from arithmetic, flat morphisms, formal schemes, infinitesimal schemes, regular rings and smoothness, relative differentials, vector fields, derivations, infinitesimal automorphisms, group schemes and basic examples, and actions of group schemes on relative schemes.
Chapter 8 gives an introduction to projective geometry, that is, to projective schemes, their special morphisms, their sheaf cohomology, and their specific geometric properties. Among other topics, the reader gets here acquainted with locally free sheaves on \(\mathbb{P}^n_A\), valuative criteria for morphisms, ample and very ample sheaves, the cohomology of quasi-coherent and coherent sheaves, the fundamental finiteness theorems for the latter, the constructions of blowing-up and contracting subschemes, base change properties, basic intersection theory, and Bertini’s theorem.
Chapter 9 is devoted to algebraic curves and the Riemann-Roch theorem. The main objects of study are here smooth projective curves over an arbitrary ground field, together with their function fields, divisors, and line bundles. The main theme in the first part of this chapter is the classical Riemann-Roch-theorem, together with the Serre duality theorem, which are treated in a very interesting, highly enlightening, and somewhat non-standard manner.
In fact, the author’s approach uses classical ideas of R. Dedekind and H. Weber [cf.: “Theorie der algebraischen Funktionen einer Veränderlichen”, Kronecker J. XCII. 181–291 (1882; JFM 14.0352.01)] and combines them effectively with the modern sheaf-theoretic and cohomological framework. The second part of this chapter discusses modern generalizations and applications of the Riemann-Roch theorem, ranging from the coarse moduli space of elliptic curves to the general idea of moduli spaces for arbitrary smooth projective curves. At the end, the general Riemann-Roch-Grothendieck theorem is explained in a more informal way, with the emphasis on some important special cases. However, the final highlight is the application to curves over finite fields, where the relationship between the Riemann-Roch theorem and the Zeta-function of such a curve is analyzed, on the one hand, and an analogue of the Riemann hypothesis (à la Mattuck-Tate and Grothendieck, 1958) is expounded on the other hand.
Chapter 10, the most advanced part of the book, describes another moduli problem in algebraic geometry, namely the representability of the Picard functor for algebraic curves and their Jacobians. Referring to the construction of the Jacobian of a compact Riemann surface as described in Chapter 5 of Volume I, the author first defines the Picard functor on the category of schemes of finite type over a base scheme B, establishes its local representability and the construction of the Picard scheme of a curve, and obtains the Jacobian of a curve as a particular Picard scheme, which turns out to be a connected projective variety equipped with the structure of a commutative group scheme, that is, an abelian variety. In the sequel, the theory of abelian varieties and their Picard schemes is developed, with Jacobians of curves as prototype, and endomorphism rings of Jacobians, the Weil pairing, Néron-Severi groups, and the ring of correspondences are treated just as well as the \(\ell\)-adic Tate modules. As the author points out in the preface to the present volume, the final goal of this book is to bring the reader to the foothills of the mountain range of étale cohomology. This is done in the last section of this concluding chapter, where an outlook to this subject is given. After explaining the basic concepts and some of the fundamental theorems, this section touches upon the famous Weil conjectures for projective varieties over a finite field, Deligne’s confirming theorem, and the special case of abelian varieties and of curves.
Finally, the study of a degenerating family of elliptic curves serves as an illustrating, highly instructive example for the nature of the Weil conjectures and for compactifications of moduli spaces, respectively.
All together, this second volume of [loc. cit.] stands out by the same features that already distinguished the foregoing Volume 1. Again, the very special and individual disposition of the subject matter, the purposeful arrangement of the material, the broad spectrum of topics and their interrelations, the great clarity of exposition, the topicality, the mathematical depth, and the reader-friendly style of writing make this text a particularly valuable enrichment of the existing course book literature in algebraic geometry and its arithmetic aspects.

MSC:

14-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry
14A05 Relevant commutative algebra
14A15 Schemes and morphisms
14F20 Étale and other Grothendieck topologies and (co)homologies
14H40 Jacobians, Prym varieties
14K05 Algebraic theory of abelian varieties
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