##
**Period mappings and period domains.**
*(English)*
Zbl 1030.14004

Cambridge Studies in Advanced Mathematics. 85. Cambridge: Cambridge University Press. xvi, 430 p. (2003).

The book under review is mainly intended as a textbook for graduate students in complex geometry. As such, its basic aim is to provide an up-to-date exposition of the theory of period mappings and period domains in modern transcendental algebraic geometry and Hermitean differential geometry.

Historically, the concept of a period of an elliptic integral can be traced back as far as to the 18th century (J. Wallis, J. Bernoulli, G. Fagnano). Later on, in the course of the 19th century, the systematic study of elliptic functions and integrals culminated in a fascinating and marvellous theory, due to the fundamental contributions by Abel, Gauss, Jacobi, Legendre, Liouville, Weierstrass, Riemann, Klein, Hurwitz, and many others. From the viewpoint of contemporary mathematics, the classical theory of elliptic integrals must be seen as the prelude to modern complex algebraic geometry and complex differential geometry, in general, and to the classification theory of complex structures (moduli problems), in particular.

With the uprise of Hodge theory in the second half of the 20th century, the theory of periods of integrals underwent a tremendous renascence. Namely, in the late 1960s, Ph. A. Griffiths began his huge program of generalizing the classical theory to higher-dimensional complex algebraic manifolds [cf.: Phillip A. Griffiths, “Selected works” Vol. 1-4, (Providence 2003; Zbl 1045.01012)], in the course of which he developed a theory of general period mappings and period domains which reflect the variation of complex structures on complex projective manifolds. Griffiths’s approach has been further elaborated in the last thirty years, and the general framework of period mappings and period domains has become a fundamental tool in the study of complex manifolds, their intrinsic geometry, analytic properties, and moduli.

The book under review, by the way dedicated to Ph. A. Griffiths, gives both a very profound introduction to the basics of Griffiths’s theory and a rich variety of recent developments and applications. According to its design, which is to provide an up-to-date text for seasoned graduate students and interested mathematical researchers, simultaneously, the book assumes the reader to be sufficiently familiar with complex-analytic and complex algebraic manifolds, basic algebraic topology, and the elements of modern differential geometry.

As to the contents of the book, the material is arranged in three principal parts as follows:

Part I: Basic theory of the period map;

Part II: The period map – algebraic methods;

Part III: Differential-geometric methods.

Each part is subdivided into about five chapters, and each chapter comes with several sections accompanied by selected working problems. To facilitate reading, the authors start each chapter with a brief outline of its contents, and three appendices at the end of the book compile some basic notions and facts from the prerequisites used throughout the text.

In part I, the first chapter is devoted to introductory examples: complex elliptic curves, Riemann surfaces of higher genus, double planes, and related Hodge structures. Chapter 2 discusses the cohomology of compact (Kähler) manifolds and the Lefschetz decomposition of cohomology groups, including the hard Lefschetz theorem. Chapter 3 is entitled “Holomorphic invariants and cohomology” {} and treats algebraically the de Rham groups, Hodge filtrations, the filtered Čech-de Rham double complex, a simplified version of hypercohomology, the cohomology of projective hypersurfaces, algebraic cycles and the statement of the Hodge conjecture, Griffiths’s intermediate Jacobian, and Abel-Jacobi maps. Chapter 4 studies the cohomology of manifolds varying in a family, in particular variations of Hodge structures, and introduces the Griffiths period domains serving as targets for period mappings. The basic properties of period maps (analyticity and horizontality) and abstract variations of Hodge structures, the central theme of the book, as well as a revision of the Abel-Jacobi map are the further topics of this chapter.

The infinitesimal study of period mappings, including infinitesimal variations of Hodge structures and the allied deformation theory of complex structures, is pursued in chapter 5, together with several instructive, concrete examples, and herewith ends the first part of the book. The following two parts are more special in nature, and significantly shorter in volume than the more spreading part I.

Part II starts with chapter 6, in which spectral sequences are used to tie up some previous loose ends. Chapter 7 turns to Koszul complexes and some applications of them to the subject of the book, including Castelnuovo’s regularity theorem, R. Donagi’s symmetrizer lemma, and a modern setting of the Noether-Lefschetz theorems in the theory of projective varieties. Chapter 8 provides further applications of Griffiths’s theory, namely various Torelli-type theorems and their significance in moduli theory. Chapter 9 introduces infinitesimal methods in order to study algebraic cycles and to explain several deep results of Ph. A. Griffiths, M. Green and C. Voisin. The basic tools developed here are normal functions, infinitesimal invariants as relative cycle classes, primitive \((p,p)\)-classes, and the Griffiths group of hypersurface sections.

Chapter 10 concludes this merely algebraic part II with further applications to algebraic cycles, focusing on M. Nori’s celebrated theorem. To this end, the authors provide the necessary account of Deligne cohomology and the theory of mixed Hodge structures as far as needed.

The final part III of the book turns to the purely differential-geometric aspects of period domains, with the main goal being to explain those curvature properties of them that are crucial for the study of period maps. In chapters 11 and 12, which are of preparatory character, the authors present several fundamental notions, techniques and results from differential geometry as necessary for this purpose, including Chern connections, principal bundles, homogeneous bundles on homogeneous spaces, canonical connections on reductive spaces, and – very important – the Lie algebra structure of groups defining period domains. Chapter 13 then gives various important applications of the basic curvature properties discussed before, among them the so-called theorem of the fixed part (on complex systems of Hodge bundles), the so-called rigidity theorem (on polarized variations of Hodge structure), and the already classical monodromy theorem of Landmann-Borel-Schmid. This chapter also discusses Higgs bundles (à la C. Simpson) and, as further applications, modern proofs of some classical theorems in complex analysis such as the Schwarz lemma and the Ahlfors lemma. Chapter 14, the last chapter of the book, gives a more general outlook of the analytical part of the theory. Namely, the authors explain the interaction between harmonic maps and Hodge theory with respect to locally symmetric spaces. This leads to important results such as the fact that compact quotients of certain period domains can never admit a Kähler metric, or that certain lattices in classical Lie groups cannot occur as the fundamental group of a Kähler manifold. Along this line, the authors touch upon the Eells-Sampson theory of harmonic maps and its connection to the more recent theory of Higgs bundles.

The three appendices at the end of the book recall some basics from (A) projective varieties and complex manifolds, (B) homology and cohomology in algebraic topology, and (C) vector bundles, connections, curvature, and Chern classes on differentiable manifolds.

The carefully compiled bibliography comprises as many as nearly 250 references, which the authors abundantly refer to throughout the text.

Apart from the numerous, skillfully selected exercises after each section, the historical remarks after each chapter, and the pointed hints for further reading, the great advantages of this masterly composed text are multifarious. The authors have touched upon various highly advanced and topical themes of central significance in contemporary geometry and physics, presented them in a just as deep-going as comprehensible manner, without discouraging the reader by deuced technicalities, and always set a high value on example-driven motivations for the many general, often abstract and involved concepts. Geometric intuition, intradisciplinary relationships, and – above all – diverse enlightening applications of period mappings and period domains are here at a premium, more than lengthy, technical and ultimately detailed proofs, and that should be very much to the benefit of the (perhaps even experienced) reader. The presentation of the vast material is very lucid and inspiring, methodologically well-planned and utmost user-friendly considering such sophisticated a complex of topics.

No doubt, together with the just as recent two-volume book “Hodge theory and complex algebraic geometry I, II” {} by C. Voisin (published in the same series “Cambridge Studies in Advanced Mathematics” {} as volumes 76 and 77 in 2002 and 2003; Zbl 1005.14002 and Zbl 1032.14002), which partly covers similar topics, the book under review is the most comprehensive and topical work in textbook form on the subject, and therefore a potential standard source in the future.

Historically, the concept of a period of an elliptic integral can be traced back as far as to the 18th century (J. Wallis, J. Bernoulli, G. Fagnano). Later on, in the course of the 19th century, the systematic study of elliptic functions and integrals culminated in a fascinating and marvellous theory, due to the fundamental contributions by Abel, Gauss, Jacobi, Legendre, Liouville, Weierstrass, Riemann, Klein, Hurwitz, and many others. From the viewpoint of contemporary mathematics, the classical theory of elliptic integrals must be seen as the prelude to modern complex algebraic geometry and complex differential geometry, in general, and to the classification theory of complex structures (moduli problems), in particular.

With the uprise of Hodge theory in the second half of the 20th century, the theory of periods of integrals underwent a tremendous renascence. Namely, in the late 1960s, Ph. A. Griffiths began his huge program of generalizing the classical theory to higher-dimensional complex algebraic manifolds [cf.: Phillip A. Griffiths, “Selected works” Vol. 1-4, (Providence 2003; Zbl 1045.01012)], in the course of which he developed a theory of general period mappings and period domains which reflect the variation of complex structures on complex projective manifolds. Griffiths’s approach has been further elaborated in the last thirty years, and the general framework of period mappings and period domains has become a fundamental tool in the study of complex manifolds, their intrinsic geometry, analytic properties, and moduli.

The book under review, by the way dedicated to Ph. A. Griffiths, gives both a very profound introduction to the basics of Griffiths’s theory and a rich variety of recent developments and applications. According to its design, which is to provide an up-to-date text for seasoned graduate students and interested mathematical researchers, simultaneously, the book assumes the reader to be sufficiently familiar with complex-analytic and complex algebraic manifolds, basic algebraic topology, and the elements of modern differential geometry.

As to the contents of the book, the material is arranged in three principal parts as follows:

Part I: Basic theory of the period map;

Part II: The period map – algebraic methods;

Part III: Differential-geometric methods.

Each part is subdivided into about five chapters, and each chapter comes with several sections accompanied by selected working problems. To facilitate reading, the authors start each chapter with a brief outline of its contents, and three appendices at the end of the book compile some basic notions and facts from the prerequisites used throughout the text.

In part I, the first chapter is devoted to introductory examples: complex elliptic curves, Riemann surfaces of higher genus, double planes, and related Hodge structures. Chapter 2 discusses the cohomology of compact (Kähler) manifolds and the Lefschetz decomposition of cohomology groups, including the hard Lefschetz theorem. Chapter 3 is entitled “Holomorphic invariants and cohomology” {} and treats algebraically the de Rham groups, Hodge filtrations, the filtered Čech-de Rham double complex, a simplified version of hypercohomology, the cohomology of projective hypersurfaces, algebraic cycles and the statement of the Hodge conjecture, Griffiths’s intermediate Jacobian, and Abel-Jacobi maps. Chapter 4 studies the cohomology of manifolds varying in a family, in particular variations of Hodge structures, and introduces the Griffiths period domains serving as targets for period mappings. The basic properties of period maps (analyticity and horizontality) and abstract variations of Hodge structures, the central theme of the book, as well as a revision of the Abel-Jacobi map are the further topics of this chapter.

The infinitesimal study of period mappings, including infinitesimal variations of Hodge structures and the allied deformation theory of complex structures, is pursued in chapter 5, together with several instructive, concrete examples, and herewith ends the first part of the book. The following two parts are more special in nature, and significantly shorter in volume than the more spreading part I.

Part II starts with chapter 6, in which spectral sequences are used to tie up some previous loose ends. Chapter 7 turns to Koszul complexes and some applications of them to the subject of the book, including Castelnuovo’s regularity theorem, R. Donagi’s symmetrizer lemma, and a modern setting of the Noether-Lefschetz theorems in the theory of projective varieties. Chapter 8 provides further applications of Griffiths’s theory, namely various Torelli-type theorems and their significance in moduli theory. Chapter 9 introduces infinitesimal methods in order to study algebraic cycles and to explain several deep results of Ph. A. Griffiths, M. Green and C. Voisin. The basic tools developed here are normal functions, infinitesimal invariants as relative cycle classes, primitive \((p,p)\)-classes, and the Griffiths group of hypersurface sections.

Chapter 10 concludes this merely algebraic part II with further applications to algebraic cycles, focusing on M. Nori’s celebrated theorem. To this end, the authors provide the necessary account of Deligne cohomology and the theory of mixed Hodge structures as far as needed.

The final part III of the book turns to the purely differential-geometric aspects of period domains, with the main goal being to explain those curvature properties of them that are crucial for the study of period maps. In chapters 11 and 12, which are of preparatory character, the authors present several fundamental notions, techniques and results from differential geometry as necessary for this purpose, including Chern connections, principal bundles, homogeneous bundles on homogeneous spaces, canonical connections on reductive spaces, and – very important – the Lie algebra structure of groups defining period domains. Chapter 13 then gives various important applications of the basic curvature properties discussed before, among them the so-called theorem of the fixed part (on complex systems of Hodge bundles), the so-called rigidity theorem (on polarized variations of Hodge structure), and the already classical monodromy theorem of Landmann-Borel-Schmid. This chapter also discusses Higgs bundles (à la C. Simpson) and, as further applications, modern proofs of some classical theorems in complex analysis such as the Schwarz lemma and the Ahlfors lemma. Chapter 14, the last chapter of the book, gives a more general outlook of the analytical part of the theory. Namely, the authors explain the interaction between harmonic maps and Hodge theory with respect to locally symmetric spaces. This leads to important results such as the fact that compact quotients of certain period domains can never admit a Kähler metric, or that certain lattices in classical Lie groups cannot occur as the fundamental group of a Kähler manifold. Along this line, the authors touch upon the Eells-Sampson theory of harmonic maps and its connection to the more recent theory of Higgs bundles.

The three appendices at the end of the book recall some basics from (A) projective varieties and complex manifolds, (B) homology and cohomology in algebraic topology, and (C) vector bundles, connections, curvature, and Chern classes on differentiable manifolds.

The carefully compiled bibliography comprises as many as nearly 250 references, which the authors abundantly refer to throughout the text.

Apart from the numerous, skillfully selected exercises after each section, the historical remarks after each chapter, and the pointed hints for further reading, the great advantages of this masterly composed text are multifarious. The authors have touched upon various highly advanced and topical themes of central significance in contemporary geometry and physics, presented them in a just as deep-going as comprehensible manner, without discouraging the reader by deuced technicalities, and always set a high value on example-driven motivations for the many general, often abstract and involved concepts. Geometric intuition, intradisciplinary relationships, and – above all – diverse enlightening applications of period mappings and period domains are here at a premium, more than lengthy, technical and ultimately detailed proofs, and that should be very much to the benefit of the (perhaps even experienced) reader. The presentation of the vast material is very lucid and inspiring, methodologically well-planned and utmost user-friendly considering such sophisticated a complex of topics.

No doubt, together with the just as recent two-volume book “Hodge theory and complex algebraic geometry I, II” {} by C. Voisin (published in the same series “Cambridge Studies in Advanced Mathematics” {} as volumes 76 and 77 in 2002 and 2003; Zbl 1005.14002 and Zbl 1032.14002), which partly covers similar topics, the book under review is the most comprehensive and topical work in textbook form on the subject, and therefore a potential standard source in the future.

Reviewer: Werner Kleinert (Berlin)

### MSC:

14C30 | Transcendental methods, Hodge theory (algebro-geometric aspects) |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

14C34 | Torelli problem |

14D07 | Variation of Hodge structures (algebro-geometric aspects) |

32G20 | Period matrices, variation of Hodge structure; degenerations |

53C43 | Differential geometric aspects of harmonic maps |