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Hodge theory and complex algebraic geometry. (Théorie de Hodge et géométrie algébrique complexe.) (French) Zbl 1032.14001
Cours Spécialisés (Paris). 10. Paris: Société Mathématique de France. viii, 595 p. (2002).
This voluminous monograph (600 pages) is devoted to the study of Hodge theory, and in particular its ubiquitous impact in transcendental algebraic geometry which is mirrored through monumental achievements by Griffiths, Deligne, Kodaira, to name a few. It consists of seven parts. The first three parts present standard materials which one can find in most textbooks in this subject; meanwhile the last four parts address an audience at some advanced level.
In order to be self-contained, part 1 consists of 4 chapters on introductory materials in several complex variables, coherent sheaves, cohomologies, etc.
Part II is dealing with the Hodge and Lefschetz decompositions on a compact complex manifold \(X\). It consists of 4 chapters: Chapter 5 is dealing with harmonic forms, Dolbeault cohomology and Serre duality. Chapter 6 presents the Hodge and Lefschetz decompositions when \(X\) admits a Kählerian structure. Chapter 7 introduces the Hodge structure and its polarizations. Chapter 8 is dealing with the logarithmic de Rham complexes which lead to the Hodge theory on a Zariski open variety.
Part III is devoted to the variations of Hodge structure. It consists of two chapters: Chapter 9 introduces the notion of variations of complex structure; in particular, the proof of the Kodaira theorem about the stability of Kählerian structure under small deformation is established. – Chapter 10 is dealing with the period domains and period mappings along the lines of P. A. Griffiths [Am. J. Math. 90, 568-626 (1968; Zbl 0169.52303) and Ann. Math. (2) 90, 460-495, 496-541 (1969; Zbl 0215.08103)].
Part IV is devoted to cohomology classes of an analytic cycle. It consists of two chapters: Chapter 11 discusses the Hodge conjectures. – Chapter 12 introduces the notion of intermediate Jacobians and establishes Griffith’s results on the Abel-Jacobi map. Also the notion of Deligne cohomology was taken up and identified as the quotient of a subgroup of the group of differential characters introduced by Cheeger and Simons.
The last three parts of this monograph present a deep intertwinning relationship between the topology of algebraic varieties, their algebraic cycles, and their Hodge structure, as well as the author’s perspective for future developments.
Part V presents the qualitative influence of Hodge theory on the topology of algebraic varieties. It consists of 4 chapters: Chapter 13 is devoted to Morse theory on affine varieties, which resulted in the proof of the Lefschetz theorem for hyperplane sections. Chapter 14 consists of the proof of the existence of Lefschetz pencils and is followed by the comparison of the primitive cohomology groups with the groups of invariant cocycles along the lines of A. Andreotti and T. Frankel [in: Global Analysis, Pap. Honor K. Kodaira, 1-20 (1969; Zbl 0191.19301)]. Chapter 15 studies the action of the monodromy on the cohomology of the fibres of a projective morphism. As an application, the description of Noether Lefschetz loci is given. Chapter 16 studies the degeneration of Leray spectral sequences. Also the notion of mixed Hodge structures is introduced, from which Deligne’s global theorem about invariant cycles results.
Part VI is devoted to the study and applications of the infinitesimal variations of the Hodge structure (IVHS) of a smooth family of projective varieties. It consists of three chapters: Chapter 17 introduces the basic ingredients in the study of IVHS and investigats the Hodge loci, which are the first line of defense to explore the Hodge conjecture. Chapter 18 is devoted to the variations of Hodge structure of complete families of sufficiently ample hypersurfaces and the algebraic properties of the Jacobian rings of projective hypersurfaces. The central theme of chapter 19 is the study of the Abel-Jacobi map of projective hypersurfaces of odd dimension. Also the notion of infinitesimal invariant of a normal function is introduced, from which a geometric description is given in the case of a normal function associated to a cycle. Chapter 20 studies in depth Nori’s connectivity theorem, a reinforced version of the Lefschetz theorem relative to a locally complete family of hypersurfaces or sufficiently ample complete intersections. Also, following Nori, the notion of a filtration on Griffith’s group is introduced and its nontriviality is established.
The last part of this monograph studies the interaction between the Chow groups of algebraic cycles and the Hodge theory. It consists of three chapters: Chapter 21 studies the fundamental properties of Chow groups, i.e. groups of algebraic cycles modulo rational equivalence. The reference here is W. Fulton’s book: “Intersection theory” (1984; Zbl 0541.14005)]. Also some classical examples of Chow groups are exhibited. Chapter 22 focuses on a Mumford theorem which says that the “finite dimensionality” of the Chow group of zero-cycles on an algebraic surface \(S\) will imply the vanishing of the geometric genus of \(S\). A generalization by Roitman indicates the strong influence of certain Chow groups of a variety \(X\) on their Hodge structure. Finally chapter 23 is devoted to the study of the converse direction of chapter 22, namely the Bloch Beilinson (BB) conjecture which predicts a complete determination of the Chow groups of an algebraic variety \(S\) with rational coefficients by their Hodge structure; a special case of this conjecture, namely when \(S\) is an algebraic surface, is the converse of the Mumford theorem which is also known as Bloch conjecture. Special cases of the Bloch conjecture were verified via the work of the Bloch and others on most algebraic surfaces. Also the computations of the Chow ring of an Abelian variety are carried out following the works of Beauville and Bloch which provide results which partially confirm the BB conjecture for Abelian varieties.
The monograph is supplemented by an extensive bibliography and a helpful series of exercises at the end of each chapter. This book is highly recommended for readers who have strong interests in this field. Certainly it should be a valuable addition into the arsenal of literature in topics in transcendental algebraic geometry.
This book is translated into English as two volumes: Part I: 2002; Zbl 1005.14002. – Part II: 2003; see the following review Zbl 1032.14002).

14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
14D07 Variation of Hodge structures (algebro-geometric aspects)
14C25 Algebraic cycles
32G20 Period matrices, variation of Hodge structure; degenerations
32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces