# Hodge Theory

This book provides a comprehensive introduction to Hodge theory, written by several authors. It is mainly aimed to graduate students but it can also be very useful to lecturers and researchers in Algebraic Geometry. It is based on lectures delivered at the 2010 Summer School on Hodge Theory at the ICTP in Trieste, Italy. In its 600 pages, it deals with the following topics:

Chapter 1. Kähler Manifolds, by Eduardo Cattani. It gives a standard introduction to complex manifolds, including the Hodge theory of harmonic forms for Kähler manifolds. It contains an appendix by Philip Griffiths with a proof of the Kähler identities using the symplectic structure of a Kähler manifold.

Chapter 2. Sheaf Cohomology, by Fouad El Zein and Loring W. Tu. It introduces sheaf cohomology making extensive use of Godement resolutions, hypercohomology, spectral sequences, and Cech cohomology. It gives a proof of the algebraic De Rham theorem and Serre's GAGA principle for projective varieties. Then it moves to the algebraic De Rham theorem in general, introducing the tool of compactifying an algebraic manifold with a divisor with normal crossings.

Chapter 3. Mixed Hogde Structures, by Fouad El Zein and Loring W. Tu. It gives a thorough exposition of the notion of Hodge structures and mixed Hodge structures, proving several relevant algebra results, including the use in spectral sequences and hypercohomology. The powerful machinery of mixed Hodge complexes is introduced to prove the theorem of Deligne on the construction of mixed Hodge structures on the cohomology of algebraic varieties. The case of smooth open varieties is done by using the logarithmic complex, and the case of singular complete varieties by using simplicial varieties.

Chapter 4. Period Domains and Period Mappings, by James Carlson. The period domain is the parametrizing space for polarised Hodge structures. For a fibration $X\to C$ with generically smooth fibers, there is a period map from $C$ to the period domain parametrizing Hodge structures of the fibers. Around the singular fibers there is a monodromy map. The cases of Hodge structures of weight $1$ and weight $2$ are addressed specifically.

Chapters 5 and 6. Hodge Theory of Maps, by Luca Migliorini and Mark Andrea de Cataldo. These deal with Hodge theory aspects associated to an algebraic map $f:X\to Y$ between two smooth projective varities. Results given include the triviality of the Leray spectral sequence for smooth projective maps, the theorem of the semisimplicity of the monodromy, the local and global invariant cycle theorems, the definition of the limit mixed Hodge structure for a singular fiber, and an introduction to intersection cohomology. Chapter 6 contains plenty of exercises.

Chapter 7. Variations of Hodge Structure, by Eduardo Cattani. It deals with the algebra associated to families of Hodge structures. For a fibration $X\to B$, the cohomology of the fibers form a local system, which is called a variation of the Hodge structure. The Kodaira-Spencer map and the Gauss-Manin connection are introduced. The situation is also studied in a general context (when the family of Hodge structures does not come from a geometric situation), and it analyses the asymptotic behaviour when we approach a singular fiber.

Chapter 8. Variations of Mixed Hodge Structure, by Patrick Brosnan and Fouad El Zein. This is a long and more technical chapter where the theory of the previous chapter is extended to the case of variations of mixed Hodge structures. Emphasis is put on the notion of infinitesimal mixed Hodge structure.

Chapter 9. Algebraic Cycles and Chow Groups, by Jacob Murre. This chapter, of more elementary nature. introduces the different Chow groups of an algebraic variety associated to algebraic cycles with different equivalences of cycles. The cycle map, intermediate Jacobians and the Deligne cohomology, are reviewed.

Chapter 10. Spreads and Algebraic Cycles, by Mark Green. For an algebraic variety $X$ defined abstractly over some field $k$, one takes an algebraic variety $S$ with $k={\mathbb{Q}}(S)$, so $X$ defines a family ${\mathcal{X}} \to S$ over the rational numbers, called the spread of $X$. The Hodge theoretical properties of this family give geometric information on the Chow groups of $X$, for the different embeddings $k\hookrightarrow {\mathbb{C}}$. This leads to the notion of absolute Hodge classes and the Bloch-Beilinson conjectures.

Chapter 11. Absolute Hodge Classes, by François Charles and Christian Schnell. Absolute Hodge classes were introduced by Deligne. They stand between the notion of Hodge class and the class of algebraic cycles. This is related to the Standard conjectures of Grothendieck and the Hodge conjecture. Absolute Hodge classes for abelian varieties are reviewed.

Chapter 12. Shimura Varieties, by Matt Kerr. The last chapter gives an introduction to symmetric domains, Shimura varieties, and CM-abelian varieties.

This is an extensive book on the topic of Hodge theory, with content ranging from basic material to very recent progress in the field. Some of the chapters are at the level of graduate students, self-contained and with a long and nice exposition. Other chapters however are addressed to a higher level mathematical audience, with a more sketchy exposition and redirecting to relevant bibliography for details.

Some of the chapters of the book are loosely tied to previous (more basic) content of the book. At some points, the chapters follow too closely the lectures in the way they were delivered. This gives the text some sense of incoherent recollection of topics.

**Submitted by Vicente Munoz |

**10 / May / 2017