1.1. From the Publisher.

This book is about Prof Claire Voisin's lectures delivered for the "Lagrange Chair" at the University 'La Sapienza' in Rome during the fall of 1996. Their subject is dominated by mirror symmetry, which gives conjecturally a new way of describing the periods of a Calabi-Yau threefold (and eventually n-fold). We first present the variations of Hodge structure of Calabi-Yau threefolds in a "mirror symmetry independent" point of view, and then mirror symmetry. The reason for doing this is that these variations of Hodge structure proved to be quite special from the point of view of differential geometry and of algebraic geometry and this is not apparent from the mirror symmetry approach. Also, it may be that some algebro-geometric aspects could play a role in a better understanding of mirror symmetry.

1.2. Review by: Bruce Hunt.
Mathematical Reviews MR1658398.

In the study of Calabi-Yau varieties, especially Calabi-Yau threefolds, a multitude of methods exists. On the one hand, there are those very fashionable methods inspired by physicists' interest in Calabi-Yaus. These include mirror symmetry and quantum cohomology. Of course there are also more traditional methods, using variations of Hodge structures, intermediate Jacobians, Abel-Jacobi maps and the like. The most conspicuous fact about the presentation of these lecture notes is that both sets of methods are described and applied to the study of Calabi-Yau threefolds.

The lecture notes consist of roughly two parts: the first consisting of general methods which are traditional in algebraic geometry, many of which were invented by Griffiths, and the second about the more popular methods resulting from the "physical side of the story". Generally speaking, Voisin herself has contributed to all aspects of the first part, and many of her original results are given an exposition. On the other hand, the second part summarizes the result of others, especially the physicists, as well as Morrison, Manin, Kontsevich and in particular Givental.

The more interesting topics studied in the first part are as follows. In Chapter 1, the contact system determined by the general Pfaffian system on the period domain is described - work of Bryant-Griffiths - the Lagrangian structure of the intermediate Jacobian fibration is discussed - work of Donagi and Markman, which characterises "quasi-horizontal" sections - and the construction of natural coordinates on the Kuranishi family is given; these latter are the famous special coordinates of the physicists, which exist by virtue of the fact that this moduli (or deformation) space is not just a Kähler manifold, but in fact a special Kähler manifold. In Chapter 2, the generic Torelli theorem for hypersurfaces is discussed - work of Donagi - and Voisin's extension of this to quintic hypersurfaces (which is one of the exceptions of Donagi's result) is sketched. In Chapter 3, the nontriviality of the Abel-Jacobi mapping for a generic deformation of a non-rigid Calabi-Yau threefold, a result of the author, is surveyed. Furthermore, using sophisticated methods of Green (Noether-Lefschetz theory), interesting cycles are constructed.

In the second part the following topics are discussed: rational curves and quantum cohomology (Chapter 4), mirror symmetry (Chapter 5) and Givental's work (Chapter 6), which solves the mirror symmetry conjecture for complete intersections.

The choice of topics is quite nice, and although the presentation is for the most part somewhat sketchy, it does give the reader a good idea of the methods involved. It is definitely recommended for those who want to get a feeling for the mathematics involved in this exciting area of "superstring motivated mathematics" without most of the technical details.

2.1. From the Publisher.

This is the English translation of Professor Voisin's book Symétrie miroir (1996)
reflecting the discovery of the mirror symmetry phenomenon. The first chapter is devoted to the geometry of Calabi-Yau manifolds, and the second describes, as motivation, the ideas from quantum field theory that led to the discovery of mirror symmetry. The other chapters deal with more specialized aspects of the subject: the work of Candelas, de la Ossa, Greene, and Parkes, based on the fact that under the mirror symmetry hypothesis, the variation of Hodge structure of a Calabi-Yau threefold determines the Gromov-Witten invariants of its mirror; Batyrev's construction, which exhibits the mirror symmetry phenomenon between hypersurfaces of toric Fano varieties, after a combinatorial classification of the latter; and, the mathematical construction of the Gromov-Witten potential, and the proof of its crucial property (that it satisfies the WDVV equation), which makes it possible to construct a flat connection underlying a variation of Hodge structure in the Calabi-Yau case. The book concludes with the first 'naive' Givental computation, which is a mysterious mathematical justification of the computation of Candelas, et al.

2.2. Review by: Bruce Hunt.
Mathematical Reviews MR1396787.

This is the first book by a mathematician on the topic of mirror symmetry, but expectations should not be too high. After all, since the advent of mirror symmetry ten years ago, it remains essentially just as conjectural as ever. For a general review of mirror symmetry up to about 1991, when a conference was held in Berkeley on this topic, see a collection of papers on this topic [Essays on mirror manifolds, Internat. Press, Hong Kong, 1992]. In this review we concentrate on the developments since then and their coverage in this book.

3.1. From the Publisher.

The first of two volumes offering a modern introduction to Kählerian geometry and Hodge structure. The book starts with basic material on complex variables, complex manifolds, holomorphic vector bundles, sheaves and cohomology theory, the latter being treated in a more theoretical way than is usual in geometry. The author then proves the Kähler identities, which leads to the hard Lefschetz theorem and the Hodge index theorem. The book culminates with the Hodge decomposition theorem. The meanings of these results are investigated in several directions. Completely self-contained, the book is ideal for students, while its content gives an account of Hodge theory and complex algebraic geometry as has been developed by P Griffiths and his school, by P Deligne, and by S Bloch. The text is complemented by exercises which provide useful results in complex algebraic geometry.

3.2. Review by: Javier A Fernández.
Mathematical Reviews MR1967689.

The purpose of the book that we review here is to provide an introduction to the Hodge theory of Kähler manifolds. This volume is a very good translation of the first part of [Théorie de Hodge et géométrie algébrique complexe, Soc. Math. France, Paris, 2002], which was based on courses taught by the author at the University of Paris 6 during 1999-2001. The second part of the translation appeared as [Hodge theory and complex algebraic geometry. II, Translated from the French by Leila Schneps, Cambridge Univ. Press, Cambridge, 2003].

The pioneering work of W V D Hodge and, especially, P Griffiths and P Deligne, established Hodge theory as an integral part of algebraic geometry characterized by the use of transcendental methods. In spite of the fact that the foundational results are decades old, the literature on Hodge theory is still very sparse and there are almost no textbooks offering a comprehensive view of it. In this respect, this book is a very valuable addition, providing a coherent introduction to many aspects of the theory that are not easy to find in printed form. It is the impression of this reviewer that this book is going to become a very common reference in this field.

Because of its introductory nature, this volume is useful for both a student trying to learn the subject as well as the researcher that can find a wealth of results in a clear and compact format. The exposition is very precise and the introduction that precedes each chapter helps the reader to focus on the main ideas of the text.

This book covers a lot of ground, in an efficient and elegant way. A student may find that some proofs have been reduced to the conceptual bare bones, making the material a bit difficult in the beginning. But this same fact will be greatly appreciated in successive readings. An aspect that could be improved to help the newcomer is the number of examples presented. A few exercises close each chapter.

The book is divided into four parts. Roughly speaking, the first part introduces the basic geometric objects, including sheaves. The second part centres on the Hodge and Lefschetz theorems. The third part deals with families, deformations and the period mapping. The fourth part introduces and studies the basic properties of Hodge classes and the Abel-Jacobi map.

3.3. Review by: Herbert Clemens.
Bulletin of the American Mathematical Society 42 (4) (2005), 507-520.

For a short extract from this review, see 4.3. below.

4.1. From the Publisher.

The 2003 second volume of this account of Kaehlerian geometry and Hodge theory starts with the topology of families of algebraic varieties. Proofs of the Lefschetz theorem on hyperplane sections, the Picard-Lefschetz study of Lefschetz pencils, and Deligne theorems on the degeneration of the Leray spectral sequence and the global invariant cycles follow. The main results of the second part are the generalised Noether-Lefschetz theorems, the generic triviality of the Abel-Jacobi maps, and most importantly Nori's connectivity theorem, which generalises the above. The last part of the book is devoted to the relationships between Hodge theory and algebraic cycles. The book concludes with the example of cycles on abelian varieties, where some results of Bloch and Beauville, for example, are expounded. The text is complemented by exercises giving useful results in complex algebraic geometry. It will be welcomed by researchers in both algebraic and differential geometry.

4.2. Review by: Javier A Fernández.
Mathematical Reviews MR1997577.

The English volume continues with the exposition of the elements of Hodge theory in the complex setting. It is the second volume of a two-volume translation of the French original cited in the heading. The first volume appeared as [Hodge theory and complex algebraic geometry. I, Translated from the French original by Leila Schneps, Cambridge University Press, Cambridge, 2002], and its format remains unchanged here.

The book is divided into three parts: the first is dedicated to the study of the topology of (families of) algebraic varieties. The second part is devoted to variations of Hodge structures, while the last part discusses algebraic cycles and their relations with Hodge theory.

The nature of this work is more advanced and specific than the first volume, making it useful for the researcher as well as the student specialising in the field. The exercises are well chosen to expand the exposition and the introduction to each chapter provides an excellent preview to the ideas and relations among the material to be exposed. The general approach is very formal and elegant, introducing powerful techniques that are essential for understanding the current research in the field.

4.3. Review of Volumes I and II by: Herbert Clemens.
Bulletin of the American Mathematical Society 42 (4) (2005), 507-520.

The task of reviewing Claire Voisin's two-volume work Hodge Theory and Complex Algebraic Geometry [V] is a daunting one, given the scope of the subject matter treated, namely, a rather complete tour of the subject from the beginning to the present, and given the break-neck pace of Voisin's clear, complete, but "take no prisoners" exposition. As is the case with most substantial mathematical treatises, digesting the content of these volumes can only occur by reconstructing and reorganising their material for oneself, led forward, of course, by the clear beacon carried by the author, one of the foremost leaders in the field. Rather than talk descriptively about the mathematics, I have chosen to exemplify below a strand of my own adventure in understanding and reconstruction as I worked my way through the two volumes with a group of graduate students. I hope this will serve as an invitation to other readers to do likewise. Their efforts will be amply rewarded.

5.1. From the Publisher.

In this book, Claire Voisin provides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. The volume is intended for both students and researchers, and not only presents a survey of the geometric methods developed in the last thirty years to understand the famous Bloch-Beilinson conjectures, but also examines recent work by Voisin. The book focuses on two central objects: the diagonal of a variety - and the partial Bloch-Srinivas type decompositions it may have depending on the size of Chow groups - as well as its small diagonal, which is the right object to consider in order to understand the ring structure on Chow groups and cohomology. An exploration of a sampling of recent works by Voisin looks at the relation, conjectured in general by Bloch and Beilinson, between the coniveau of general complete intersections and their Chow groups and a very particular property satisfied by the Chow ring of K3 surfaces and conjecturally by hyper-Kähler manifolds. In particular, the book delves into arguments originating in Nori's work that have been further developed by others.

5.2. Review by: Chris A M Peters.
Mathematical Reviews MR3186044.

This book is based on two lecture series by the author: the Rademacher lectures at the University of Pennsylvania, Philadelphia (2011-2012), and the Herman Weyl lectures at the Institute for Advanced Study, Princeton (2011). There are two main themes around which the book revolves: the first is the Bloch-Srinivas principle of the decomposition of the diagonal and its variants, and the second is the method of spreading cycles and the repercussions on the topology of algebraic varieties. These two intertwined themes have also been guiding principles in one of the major strands of the author's research, and the lectures as well as this book introduce the interested mathematician to this research.

This is not an easy subject since it requires thorough knowledge of the complex analytic approach to algebraic geometry, notably Hodge theory, as well as the more algebraic point of view, e.g., the theory of Chow groups and related Chow motives. Clearly it is a rather impossible task to explain this background as well as the many research articles on which the lectures are based in a mere 163 pages. Choices had to be made: in particular, many foundational things had to be left out. Nowadays one can, however, easily find these in the existing literature, e.g., in the author's set of monographs [Hodge theory and complex algebraic geometry. I, translated from the French original by Leila Schneps, Cambridge Stud. Adv. Math., 76, Cambridge Univ. Press, Cambridge, 2002; Hodge theory and complex algebraic geometry. II, translated from the French by Leila Schneps, Cambridge Stud. Adv. Math., 77, Cambridge Univ. Press, Cambridge, 2003].
...
... the book contains many interesting results on algebraic cycles, most of which have not been presented in book form before. Many complete proofs are given, mostly just as they appear in the original research articles, but accompanied with helpful comments and side remarks. Although the lectures on which the book is based were meant for a larger audience, it is only fair to say that the book itself is more directed towards the researcher in algebraic geometry who, moreover, is well acquainted with Hodge theory. Such a reader will find a rich collection of ideas as well as detailed machinery with which to attack difficult problems in the field. Any complex geometer interested in the interplay between algebraic cycles, Hodge theory and algebraic topology should have this book on his or her shelf.

6.1. From the Publisher.

What is mathematical knowledge? What is a mathematical theory for? And what is mathematics? The nature and purpose of mathematics remain mysterious, and it often appears very abstract.

Mathematics, however, has a well-defined notion of truth: what is demonstrated is true. For the purposes of demonstration, precisely, mathematics uses tools. Language, first of all, plays a fundamental role in the development of the definition, the hypothesis, the demonstration and the theorem. Mathematics also has a close link with logic, so much so that one might wonder whether they should be distinguished. In a diametrically opposed way, we can question the place of geometry in modern research in mathematics.

In this short essay, Claire Voisin tells, from the inside, how mathematics is done, and shows us that abstraction is not complexity but that on the contrary it is born from the constant concern for simplification and economy of thought. which characterises mathematics.

6.2. From the Introduction.

To start; it is perhaps good to question mathematical knowledge and the position that mathematics occupies among the other sciences. First of all, the term "science" does not apply in an obvious way because, etymologically, it means "to know"; however, the very nature and object of mathematical knowledge are mysterious. A specific feature of mathematics is the fact that it works hand-in-hand with language, which plays a fundamental role and appears in the organisation of the mathematical process at all stages: the definition, the hypothesis, the demonstration and the theorem.

The definition makes it possible to summon objects using a formalism. If I start a mathematical statement, I first give myself objects. It is a verbal statement which takes the form: "Let G be a group ..." (for example). I am carrying out two operations of a linguistic nature here. I first give a name: G (capital letter because it is a group). Mathematicians are extremely particular about questions of notation because they have to name their objects, and to do this they use letters taken from various alphabets, which will serve as a sort of proper names. But you have to be able to find your way; and these letters therefore often have specific connotations, according to the alphabet for example. It is a first form of relationship with language.

I then speak of a "group", an object which has been the subject of a definition, or even a prior theory. When I say: "Let G be a group ...", not only do I name, but I also call up the entirety of a mathematical object and its attributes, that is to say a whole set of axioms and of properties which have been the subject of a prior theory.

A specificity of mathematical knowledge is that mathematicians are the only scientists to have a well-defined notion of "true". For a mathematician, a statement that is true is a statement that is demonstrated. Unfortunately, this also says that mathematical knowledge, assuming that this has a meaning, is conditional knowledge, that is to say conditioned on hypotheses. A mathematician does not say "I know", but "I know that such and such a hypothesis leads to such and such a conclusion". He does not does not just say "I know", he demonstrates it. Thus, one of the key words in mathematics is "demonstration". We are therefore no longer only in the register of knowledge, in the sense of an encyclopaedia, a list official expression of things we know, but also in the register of doing. We say: "to demonstrate."

Language, and formal language, are therefore very important in mathematics. This is what all our activity, all our thinking is based on. Paradoxically, alongside this extreme formalism, certain fundamental mathematical objects, which can be considered by non-mathematicians as abstract, appear to mathematicians with a very crude reality, as if they were more real than the reality that surrounds us. Any mathematician can answer the question "what mathematical object would you take to a desert island?" For my part, this would most certainly be a Hodge structure.

6.3. Review by: Nicolas Michel.
Mathematical Reviews MR4321539.

In this short booklet, Claire Voisin reflects on her experience as a mathematician and on her conception of algebraic geometry as a science. This book is part of a collection of essays in which recipients of the CNRS Gold Medal (arguably the highest scientific research distinction in France, awarded once every year) are offered an opportunity to communicate to a broad audience the core ideas and principles of their work. Consequently, while this book frequently refers to advanced concepts and objects (parallel transport, Hodge bundles, etc.), the emphasis is placed on general issues, which should concern a wide range of students, philosophers, and amateurs of mathematics.

How do mathematicians wield language, words, and symbols to construct increasingly abstract structures and to rigorously prove theorems about them? What constitutes a mathematical theory, and how do mathematicians use them? What even is (a) "geometry" in the 21st century, and what sorts of tasks motivate mathematical research about geometrical objects? Those are the essential questions to which Voisin seeks to provide elements of an answer. The result is a swift journey across a variety of concepts - some technical, like o-minimality or invariance; others more philosophical in nature, like abstraction or unification. The brevity of Voisin's prose will certainly leave more interrogations open than most readers will have started with, but it will nonetheless serve as a stimulating invitation to engage with the more rewarding aspects of modern algebraic geometry. Furthermore, this book's appeal is not restricted to its intended audience. Indeed, to historians and philosophers in search of current-day echoes of various epistemological themes in relation to questions of mathematical truth, language, and practice, it will provide a valuable document testifying to how a leading mathematician views her own field nowadays.