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Complex geometry. An introduction. (English) Zbl 1055.14001
Universitext. Berlin: Springer (ISBN 3-540-21290-6/pbk). xii, 309 p. (2005).
The book under review provides an introduction to the contemporary theory of compact complex manifolds, with a particular emphasis on Kähler manifolds in their various aspects and applications. As the author points out in the preface, the text is based on a two-semester course taught in 2001/2002 at the University of Cologne, Germany. Having been designed for third-year students, the aim of the course was to acquaint beginners in the field with some basic concepts, fundamental techniques, and important results in the theory of compact complex manifolds, without being neither too basic nor too sketchy. Also, as complex geometry has undergone tremendous developments during the past five decades, and become an indispensable framework in modern mathematical physics, the author has tried to teach the subject in such a way that would enable the students to understand the more recent developments in the field, too, up to some of the fascinating aspects of the stunning interplay between complex geometry and quantum field theory in theoretical physics.
The present text, as an outgrowth of this special course in complex geometry, does evidently reflect these emphatic intentions of the author’s in a masterly manner. Keeping the prerequisites from complex analysis and differential geometry to an absolute minimum, he provides a streamlined introduction to the theory of compact complex manifolds and Kählerian geometry, with many outlooks and applications, but without trying to be encyclopedic or panoramic. As to the precise contents, the text consists of six chapters and two appendices.
Chapter 1 provides as much of local complex-analytic geometry as needed for the global study of complex manifolds, including the basics of the theory of holomorphic functions of several variables, complex and Hermitian structures and the calculus of differential forms. Chapter 2 discusses general complex manifolds, their holomorphic vector bundles, divisors and line bundles, and those basic techniques like projective embeddings, blow-ups, and the differential calculus on complex manifolds, including almost complex structures and Dolbeault cohomology. Together with many concrete examples, the reader makes here his first encounters with the algebraic-geometric aspects of certain compact complex manifolds.
Chapter 3 is devoted to (mostly compact) Kähler manifolds, their Hodge theory, cohomology, and the related Lefschetz theorems. This chapter comes with three appendices complementing the basic theory of Kähler manifolds in three directions. The first appendix gives a homological interpretation of the fundamental \(\partial\overline\partial\)-lemma in terms of the differential graded algebra of a compact Kähler manifold, showing that the latter is formal, and the second appendix is virtually a first introduction to super Lie algebras in Kählerian geometry. Together with a brief outlook to hyperkähler manifolds, this appendix points to some mathematical aspects of supersymmetry in mathematical physics. Finally, the third appendix touches upon the more general theory of weighted rational Hodge structures as a generalization of the Hodge theory of Kähler manifolds.
Chapter 4 turns to further indispensable tools in complex geometry and Hermitian differential geometry such as general holomorphic vector bundles, Hermitian bundles, Serre duality, connections on vector bundles, curvature forms, and Chern classes. In the two appendices to this chapter, the author illustrates the interplay of complex geometry and Riemannian differential geometry by discussing the relation between the Levi-Civita connection and the Chern connection on a Kähler manifold, the holonomy on complex manifolds, Hermite-Einstein metrics, and Kähler-Einstein metrics.
Chapter 5 is to demonstrate the ubiquity and power of cohomological methods in complex algebraic geometry by means of three central, classical results. Those are the Hirzebruch-Riemann-Roch theorem, the Kodaira vanishing theorem, and the Kodaira embedding theorem. Except for the Hirzebruch-Riemann-Roch theorem, complete proofs of everything else in this chapter are given, together with various important and illustrating applications of these crucial results.
Chapter 6 serves as a first introduction to the deformation theory of complex structures. In the first section, the author shows how to study deformations by power series expansions, which leads to the Maurer-Cartan equation. This approach is taken with a view towards the moduli of Calabi-Yau manifolds and the mirror symmetry phenomenon, because it can be carried out, in this context, by applying the more recent Tian-Todorov lemma and its generalization. All the facts used in this section are proved in full detail, whereas the subsequent section merely surveys the more abstract parts of deformation theory à la Ehresmann, Kodaira-Spencer, and Kuranishi. In an appendix to this last chapter, the author introduces abstract and concrete differential Gerstenhaber-Batalin-Vilkovisky algebras (dGBV-algebras) and their deformations. This highly useful homological framework is used to interpret the contents of Section 6.1 from another angle, which recently has turned out to be very efficient in the construction of Frobenius manifolds and in the formulation of mirror symmetry.
The author has added two general appendices at the end of the book. Those are meant to help the unexperienced reader to recall a few basic concepts and facts from differential geometry, Hodge theory on differentiable manifolds, sheaf theory, and sheaf cohomology. This very user-friendly service makes the entire introductory text more comfortable for less seasoned students, perhaps also for interested and mathematically less experienced physicists, although the author does not claim absolute self-containedness of the book. The entire text comes with a wealth of enlightening examples, historical remarks, comments and hints for further reading, outlooks to other directions of research, and numerous exercises after each section. The exercises are far from being bland and often quite demanding, but they should be mastered by ambitious and attentive readers, in the last resort after additional reading. Finally, there is a very rich bibliography of 118 references, also from the very recent research literature, which the author profusely refers to throughout the entire text. The whole exposition captivates by its clarity, profundity, versality, and didactical strategy, which lead the reader right to the more advanced literature in complex geometry as well as to the forefront of research in geometry and its applications to mathematical physics. No doubt, this book is an outstanding introduction to modern complex geometry.

14-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
32-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to several complex variables and analytic spaces
32J27 Compact Kähler manifolds: generalizations, classification
32G05 Deformations of complex structures
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