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Introduction to Hodge theory. (Introduction à la théorie de Hodge.) (French. English summary) Zbl 0849.14002

Panoramas et Synthèses. 3. Paris: Société Mathématique de France. vi, 272 p. (1996).
The origin of what is currently meant by the notion of Hodge theory can be traced back to W. V. D. Hodge’s fundamental work accomplished in the 1930s. In modern terminology, Hodge prepared the ground for describing the De Rham cohomology algebra of a Riemannian manifold in terms of its harmonic differential forms. In the following two decades, Hodge’s decomposition principle has been extended to the (then) new sheaf-theoretic and cohomological framework of Hermitean differential geometry, complex-analytic geometry, and transcendental algebraic geometry. The names of G. De Rham, A. Weil, K. Kodaira, and many others stand for the tremendous progress achieved during this period, in particular with regard to deformation and classification theory in these areas. The special algebraic structures (Hodge structures) arising from Hodge decompositions and their generalizations have led to a rather independent field of research in geometry, precisely to the so-called Hodge theory, which represents a powerful and indispensible toolkit for contemporary complex geometry, general algebraic geometry, and – nowadays – also for mathematical physics. The vast activity in Hodge theory and its related areas, especially during the recent twenty years, is not reflected in the current textbook literature, at least not comprehensively or in an updated form compiling the various recent aspects and applications, so that a panoramic overview of the present state of art must be regarded as a highly welcome (and needed) service to the mathematical community.
A conference on the present state of Hodge theory, serving exactly that purpose, took place at the University of Grenoble (France) in November 1994. The book under review grew out of the series of lectures which the authors delivered at this meeting. The aim of the text is to develop a number of fundamental concepts and results of classical and modern Hodge theory, and in this the book is prepared for students and non-expert researchers in the field, who wish to get acquainted in depth with the subject, and obtain a profound up-to-date knowledge of its present level of development. – The material is divided into three main parts, each of which is written by different authors and devoted to various central and complementary aspects of the theory.
Part I, written by J.-P. Demailly, is entitled “\(L^2\)-Hodge theory and vanishing theorems”. The author discusses in detail two fundamental applications of Hilbert \(L^2\)-space methods to complex analysis and algebraic geometry, respectively. This part adopts basically the analytic viewpoint and consists, on its side, of two chapters. Chapter 1 provides an introduction to standard complex Hodge theory, including the basics on Hermitean and Kähler geometry, differential operators on vector bundles, Hodge decomposition, Hodge degeneration, the spectral sequence of Hodge-Frölicher, Gauss-Manin connexion, and the deformation behavior of the Hodge groups (after Kodaira). Chapter 2 is devoted to \(L^2\)-estimates for the \(\overline \partial \)-operator and the resulting vanishing theorems for cohomology groups of Kähler manifolds and projective varieties. The main topics here are the classical methods of Oka, Bochner, and Hörmander in pseudo-convex analysis, their consequences for cohomology vanishing, as well as the more recent but already well-known fundamental contributions by the author himself towards the interpretation of the great vanishing theorems of A. Nadel and of Kawamata-Viehweg. – The concluding two sections of this chapter deal with the property of very-ampleness of line bundles on projective varieties. The first central result discussed here is the author’s analytic approach to the famous conjecture of Fujita, culminating in an improvement of Y.-T. Siu’s very recent theorem on an effective bound for very-ampleness [cf. “Effective very ampleness”, Invent. Math. 124, No. 1-3, 563-571 (1996)]. The second central result is an effective version of the classical “Big embedding theorem of Matsusaka”, whose surprisingly simple proof is due to the author himself (1996), based on some foregoing work of Y.-T. Siu [Ann. Inst. Fourier 43, No. 5, 1387-1405 (1993; Zbl 0803.32017)], and methodically related to the effective bound for very-ampleness discussed before. These two last sections provide a particularly up-to-data account on the newest developments in analytical Hodge theory and its (algebraic) applications.
Part II of the text, written by L. Illusie, is entitled “Frobenius and Hodge degeneration”. These notes aim at introducing non-specialists to those methods and techniques of algebraic geometry over a field of characteristic \(p > 0\), which have been used by P. Deligne and the author to give an algebraic proof of the Hodge degeneration and the Kodaira-Akizuki-Nakano vanishing theorem for smooth projective varieties in characteristic zero. Basically, this part of the book is a careful, detailed introduction to the important work “Relèvements modulo \(p^2\) et décomposition du complexe de De Rham” [Invent. Math. 89, 247-270 (1987; Zbl 0632.14017)] by P. Deligne and L. Illusie. Here the reader is assumed to bring along some basic knowledge of the theory of algebraic schemes and of homological algebra (in categories). After recalling the basics on schemes, differentials and the algebraic De Rham complex in characteristic \(p>0\), the author discusses the following topics: smoothness and coverings, the Frobenius morphism and the Cartier isomorphism, derived categories and spectral sequences, decomposition theorems, vanishing theorems in characteristic \(p\), degeneration theorems, the standard techniques for passing from characteristic \(p\) to characteristic zero, and the proof of the above mentioned degeneration and vanishing theorems. The concluding section of this part points to some recent developments and open problems concerning Hodge theory in characteristic \(p\).
Also this part is essentially self-contained, and most proofs are given in detail. Some proofs are – quite naturally – at least outlined, assuming the reader to follow the precise hints to the related textbook literature (mostly EGA) and original papers.
Part III of the book, written by J. Bertin and C. Peters, is entitled “Variations of Hodge structures, Calabi-Yau manifolds, and mirror symmetry”. It consists again of two main chapters, whose interrelation is beautifully explained in a comprehensive introduction. – Chapter I is devoted to the comparatively elementary part of the theory of variation of Hodge structures and its applications in complex algebraic geometry. This includes detailed descriptions of the Hodge bundles, the Hodge filtrations, the De Rham cohomology sheaves, the Gauss-Manin connexion in its general setting (after Katz and Oda) and with its transversality property (due to Griffiths), variations and infinitesimal variations of Hodge structures, the Griffiths period domains for polarized Hodge structures, mixed Hodge structures, limits of Hodge structures (after Deligne), the Picard-Lefschetz theory and the local monodromy theorem, Deligne’s degeneration criteria for Hodge spectral sequences, and a brief discussion of the method of vanishing cycles. At the end, the authors give a sketch of the use of Higgs bundles for the construction of variations of Hodge structures, mainly by following Simpson’s approach [cf.: C. T. Simpson, Proc. Int. Congr. Math., Kyoto 1990, Vol. I, 747-756 (1991; Zbl 0765.14005)], as well as some comments on M. Saito’s work on Hodge modules, intersection cohomology, and \({\mathcal D}\)-modules in algebraic analysis. – Chapter II reflects the fact that Calabi-Yau manifolds, their Hodge theory, and their mirror symmetry have recently gained enormous significance in both algebraic geometry and theoretical physics, particularly in constructing two-dimensional conformal quantum field theories. The material presented here covers the fundamental facts on Calabi-Yau manifolds, their construction and deformation theory, and their mirror properties. After a digression on the cohomology of hypersurfaces (after Griffiths and Dimca), which is used for the description of the link between the Picard-Fuchs equation and the variation of Calabi-Yau structures, the variation of Hodge structures for families of Calabi-Yau threefolds, their Yukawa couplings, and their mirror symmetries are explained in more depth. The interested reader can find a very complete and comprehensive account on this subject in the recent monography “Symétrie miroir” by C. Voisin [Panoramas et Synthèses, No. 2 (1996; see the preceding review)]. In a concluding section, the authors discuss (following an idea of P. Deligne) a possible approach to mirror symmetry via a certain duality between variations of Hodge structures for Calabi-Yau threefolds. A rich bibliography enhances this very systematic and lucid treatise.
Altogether, the present book, in all its three parts, which consistently refer to each other, may be regarded as a masterly introduction to Hodge theory in its classical and very recent, analytic and algebraic aspects. Aimed to students and non-specialists, it is by far much more than only an introduction to the subject. The material leads the reader to the forefront of research in many areas related to Hodge theory, and that in a detailed and highly self-contained manner. As such, this text is also a valuable source for active researchers and teachers in the field, in particular due to the utmost carefully arranged index at the end of the book.

MSC:

14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14F17 Vanishing theorems in algebraic geometry
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
14D07 Variation of Hodge structures (algebro-geometric aspects)
13A35 Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure
58A14 Hodge theory in global analysis
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
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