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Hodge theory and complex algebraic geometry. II. Transl. from the French by Leila Schneps. (English) Zbl 1032.14002
Cambridge Studies in Advanced Mathematics. 77. Cambridge: Cambridge University Press. ix, 351 p. £60.00; $ 85.00 k (2003).
This book is the second volume of a comprehensive introduction to the methods of modern transcendental algebraic geometry. The one-volume French original of the whole treatise appeared in 2002 (see the preceding review Zbl 1032.14001), published by Société Mathématique de France, and presented the contents of a course taught by the author at Université de Paris VI in 1998-1999 and 1999-2000 within a DEA program for advanced students. The first volume of the English translation (2002; Zbl 1005.14002) provided the fundamental results of the underlying Kählerian geometry and Hodge theory. The main topics presented in this first volume were, among others, the Hodge decomposition of compact Kähler-manifolds, the Lefschetz decomposition, mixed Hodge structures and their variations, the period map, cycle classes, Deligne cohomology, and the Abel-Jacobi map. The second volume, now under review, stands at a more advanced level, in that it turns to the description of much finer interactions between the topology of algebraic varieties, their algebraic cycles and their Hodge theory. Two central themes which dominate this second volume are the various Lefschetz theorems and Leray spectral sequences, together with their powerful applications to the study of complex projective varieties.
According to the aim of this second volume, the text is separated into three main parts, each of which comes with several chapters, sections and subsections.
Part I is entitled “The topology of algebraic varieties” and contains the first four chapters of this volume. Chapter 1 is devoted to the Lefschetz theorem on hyperplane sections, which is proved by using Morse theory on affine varieties. Chapter 2 discusses Lefschetz pencils and gives an alternative, more complex-analytic approach to the results of the preceding chapter. Chapter 3 studies the monodromy action on the cohomology of the fibres of a projective morphism, thereby exhibiting the monodromy representation as an essential ingredient of the theory of variations of Hodge structure (à la Ph. Griffiths). As an application, the Noether-Lefschetz theorem is also explained. Chapter 4 deals with the fundamental Leray spectral sequence and illustrates, along this path, the consequences of Hodge theory for the study of the topology of families of algebraic varieties, ending up with Deligne’s theorem and the invariant cycles theorem.
Part II is entitled “Variations of Hodge structure” and is formed by chapters 5-8 of this volume. The main topic discussed here is the study of infinitesimal variations of Hodge structure for a family of smooth projective varieties, together with special applications to the case of complete families of hypersurfaces or complete intersections, respectively. Chapter 5 introduces the main objects in the theory of variations of Hodge structure, with particular emphasis of the Leray spectral sequence in this context. The following three chapters are then devoted to a closer study of them in special cases. Chapter 6 deals with variation of Hodge structure of complete families of sufficiently ample projective hypersurfaces, concluding with the according generic Torelli theorem, whilst chapter 7 turns to normal functions and infinitesimal invariants, including the Jacobi fibration, a deeper study of the Abel-Jacobi map and the special case of hypersurfaces of high degree in projective space. Chapter 8, the highlight of this part of the book, presents the proof of the connectivity theorem by M. Nori [Invent. Math. 111, 349-373 (1993; Zbl 0822.14008)], which can be seen as a strengthened Lefschetz theorem for locally complete families of sufficiently ample hypersurfaces or complete intersections. As the author points out, this result brings the reader back to the starting point of this second volume, namely the Lefschetz theorem on hyperplane sections, and enables us to appreciate the immense power of the theory of infinitesimal variations of Hodge structure in complex algebraic geometry.
Part III, the final part of the entire treatise, is closely linked to the preceding part and entitled “Algebraic cycles”. This final part contains the remaining three chapters and is devoted to exploring the interactions between the Hodge theory and the algebraic cycles of a projective algebraic variety. Chapter 9 introduces the Chow groups of a variety, the intersection and cycle classes, and gives many important concrete examples of them. The general theme of chapter 10 is then the interrelation between the complexity of the Chow groups and the complexity of the Hodge structures of a smooth complex projective variety, culminating in the proof of Mumford’s theorem on the kernel of the Abel-Jacobi map for zero-cycles on a smooth surface [D. Mumford, J. Math. Kyoto Univ. 9, 195-204 (1969; Zbl 0184.46603)]. The related results of A. A. Roĭtman [Mat. Sb., N. Ser. 89 (131), 569-585 (1972; Zbl 0259.14003), Mat. Zametki 28, 85-90 (1980; Zbl 0457.14007) and Ann. Math. (2) 111, 553-569 (1980; Zbl 0504.14006)] are are also touched upon, in this context. – The final chapter 11 assembles the evidence in favour of the conjectural converse to Mumford’s theorem, the so-called “Bloch conjecture”, and its various generalizations. The first section deals with the case of a smooth surface \(S\) satisfying \(p_g:= \dim_\mathbb{C} H^{2,0}(S)=0\), the second section discusses the so-called generalized Bloch conjecture, and the concluding third section is devoted to the Bloch-Beilinson conjecture for Abelian varieties. The author concludes this part with some explicit computations in the Chow ring of an Abelian variety, following A. Beauville and S. Bloch, which provide structure results that partially confirm the Bloch-Beilinson conjecture in this special case.
As in the first volume of this English translation of C. Voisin’s lecture notes, each chapter comes with a set of carefully selected, accompanying exercises, which lead the reader into related, often even much deeper grounds. In addition, this second volume has been given its own introduction. Very much to the benefit of the reader, the author recalls here the central results of the first volume, which are constantly used throughout this second volume too, and she gives a synthetic picture of the interactions between the three main topics discussed here, so to speak as a strategic guideline, so that the reader will not get confused by the separation of this volume into three seemingly independent parts.
All together, the author has maintained her masterly style also throughout this second, much more advanced volume, just as expected. The entire two-volume text is highly instructive, inspiring, reader-friendly and generally outstanding. Without any doubt, these two volumes must be seen as an indispensible standard text on transcendental algebraic geometry for advanced students, teachers, and also researchers in this contemporary field of mathematics. The author provides, simultaneously and in a unique manner, both a complete didactic exposition and an up-to-date presentation of the subject, which is still a rather exceptional feature in the textbook literature.

MSC:
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces
14C25 Algebraic cycles
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
32G20 Period matrices, variation of Hodge structure; degenerations
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A characterisation of $\mathbb{P}^n$
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