# Hodge Theory (MN-49)

Eduardo Cattani
Phillip A. Griffiths
Lê Dũng Tráng
Pages: 608
https://www.jstor.org/stable/j.ctt6wpzdg

1. Front Matter
(pp. i-iv)
2. Contributors
(pp. v-vi)
(pp. vii-xiv)
4. Preface
(pp. xv-xviii)
5. Chapter One Introduction to Kähler Manifolds
(pp. 1-69)
Eduardo Cattani

This chapter is intended to provide an introduction to the basic results on the topology of compact Kähler manifolds that underlie and motivate Hodge theory. Although we have tried to define carefully the main objects of study, we often refer to the literature for proofs of the main results. We are fortunate in that there are several excellent books on this subject and we have freely drawn from them in the preparation of these notes, which make no claim of originality. The classical references remain the pioneering books by Weil [34], Chern [6], Morrow and Kodaira [17, 19], Wells [35],...

6. Chapter Two From Sheaf Cohomology to the Algebraic de Rham Theorem
(pp. 70-122)
Fouad El Zein and Loring W. Tu

The concepts of homology and cohomology trace their origin to the work of Poincaré in the late nineteenth century. They attach to a topological space algebraic structures such as groups or rings that are topological invariants of the space. There are actually many different theories, for example, simplicial, singular, and de Rham theories. In 1931, Georges de Rham proved a conjecture of Poincaré on a relationship between cycles and smooth differential forms, which establishes for a smooth manifold an isomorphism between singular cohomology with real coefficients and de Rham cohomology.

More precisely, by integrating smooth forms over singular chains on...

7. Chapter Three Mixed Hodge Structures
(pp. 123-216)
Fouad El Zein and Lê Dũng Tráng

We assume that the reader is familiar with the basic theory of manifolds, basic algebraic geometry as well as cohomology theory. For instance, we shall refer to Chapter 1 and Chapter 2 in this book and we recommend to the reader books like e.g., [45], [3], or [40], and the beginning of [28] and, for complementary reading on complex algebraic and analytic geometry, [43, 39, 47].

According to Deligne, the cohomology spaceHn(X, C) of a complex algebraic varietyXcarries two finite filtrations by complex subvector spaces: the rationally defined weight filtrationWand the Hodge filtrationF...

8. Chapter Four Period Domains and Period Mappings
(pp. 217-256)
James Carlson

The aim of these lectures is to develop a working understanding of the notions of period domain and period mapping, as well as familiarity with basic examples thereof. The fundamental references are [10] and [11]. We will not give specific references to these, but essentially everything below is contained in, or derivative of these articles. Three general references are [12], [15], and [5].

In previous lectures you have studied the notion of a polarized Hodge structureHof weightnover the integers, for which the motivating example is the primitive cohomology in dimensionnof a projective algebraic manifold...

9. Chapter Five The Hodge Theory of Maps
(pp. 257-272)
Mark Andrea de Cataldo and Luca Migliorini

These three lectures summarize classical results of Hodge theory concerning algebraic maps. Lectures 4 and 5, to be delivered by M. A. de Cataldo, will discuss more recent results. I will not try to trace the history of the subject nor attribute the results discussed. Coherently with this policy, the bibliography only contains textbooks and a survey, and no original papers. Furthermore, quite often the results will not be presented in their maximal generality; in particular, I’ll always stick to projective maps, even though some results discussed hold more generally.

Hodge theory gives nontrivial restrictions on the topology of a...

10. Chapter Six The Hodge Theory of Maps
(pp. 273-296)
Mark Andrea de Cataldo and Luca Migliorini

These are the lecture notes from my two lectures 4 and 5. To get an idea of what you will find in them, parse the table of contents at the beginning of the book. The lectures had a very informal flavor to them and, by choice, the notes reflect this fact. There are plenty of exercises and some references so you can start looking things up on your own. My book [5] contains some of the notions discussed here, as well as some amplifications.

1. We say that a sheaf of abelian groupsIon a topological spaceXis...

11. Chapter Seven Introduction to Variations of Hodge Structure
(pp. 297-332)
Eduardo Cattani

The modern theory of variations of Hodge structure (although some authors have referred to this period as the prehistory) begins with the work of Griffiths [21, 22, 23] and continues with that of Deligne [16, 17, 18] and Schmid [35]. The basic objects of study areperiod domainswhich parametrize the possible polarized Hodge structures in the cohomology of a given smooth projective variety. An analytic family of such varieties gives rise to a holomorphic map with values in a period domain, satisfying an additional system of differential equations. Moreover, period domains are homogeneous quasi-projective varieties and, following Griffiths and...

12. Chapter Eight Variations of Mixed Hodge Structure
(pp. 333-409)
Patrick Brosnan and Fouad El Zein

The object of the paper is to discuss the definition of admissible variations of mixed Hodge structure (VMHS), the results of Kashiwara in [22] and applications to the proof of algebraicity of the locus of Hodge cycles [4, 5]. Since we present an expository article, we explain the evolution of ideas starting from the geometric properties of algebraic families as they degenerate and acquire singularities.

The study of morphisms in algebraic geometry is at the origin of the theory of VMHS. Letf:XVbe a smooth proper morphism of complex algebraic varieties. By Ehresmann’s theorem, the...

13. Chapter Nine Lectures on Algebraic Cycles and Chow Groups
(pp. 410-448)
Jacob Murre

These are the notes of my lectures in the ICTP Summer School and conference on “Hodge Theory and Related Topics” in 2010.

The notes are informal and close to the lectures themselves. As much as possible I have concentrated on the main results. Especially in the proofs I have tried to outline the main ideas and mostly omitted the technical details. In order not to “wave hands” I have often written “outline or indication of proof” instead of “proof”; on the other hand when possible I have given references where the interested reader can find the details for a full...

14. Chapter Ten The Spread Philosophy in the Study of Algebraic Cycles
(pp. 449-468)
Mark L. Green

There are two ways of looking at a smooth projective variety in characteristic 0:

Geometric:Xis a compact Kähler manifold plus a Hodge class, embedded in CPNso that the hyperplane bundleHpulls back to a multiple of the Hodge class.

Algebraic:Xis defined by homogeneous polynomials in$k[{x_0}{\rm{,}}...{\rm{,}}{x_N}]$for a fieldkof characteristic 0. We may takekto be the field generated by ratios of coefficients of the defining equations ofX, and hence we may takekto be finitely generated over Q. There are algebraic equations with coefficients in Q that, applied...

15. Chapter Eleven Notes on Absolute Hodge Classes
(pp. 469-530)
François Charles and Christian Schnell

Absolute Hodge classes first appear in Deligne’s proof of the Weil conjectures for K3 surfaces in [14] and are explicitly introduced in [16]. The notion of absolute Hodge classes in the singular cohomology of a smooth projective variety stands between that of Hodge classes and classes of algebraic cycles. While it is not known whether absolute Hodge classes are algebraic, their definition is both of an analytic and arithmetic nature.

The paper [14] contains one of the first appearances of the notion of motives, and is among the first unconditional applications of motivic ideas. Part of the importance of the...

16. Chapter Twelve Shimura Varieties: A Hodge-Theoretic Perspective
(pp. 531-576)
Matt Kerr

In algebraic geometry there is a plethora of objects whichturn outby big theorems to be algebraic, but which aredefinedanalytically:

projective varieties, as well as functions and forms on them (by Chow’s theorem or GAGA [Serre1956]);

Hodge loci, and zero loci of normal functions (work of Cattani–Deligne–Kaplan [CDK1995] and Brosnan–Pearlstein [BP2009]);

complex tori with a polarization (using theta functions, or using the embedding theorem [Kodaira1954]);

Hodge classes (if a certain \$1,000,000 problem could be solved) and of concern to us presently,

modular (locally symmetric) varieties which can be thought of as the Γ\D’s for...

17. Index
(pp. 577-589)