Chapter 3 documents how the meaning of fake news changes over time. I look at how fake news was understood during the 2000s and 2010s – prior to the Trump era – in the New York Times. I discuss how fake news was primarily understood in three ways – related to fabricated stories passed off as real news events, as entertainment content pertaining to current events, and as government propaganda masquerading as journalism. Further, the US media’s understanding of fake news has shifted over the years. My analysis of the New York Times’ coverage of fake news in the 2010s finds that the paper defined the concept in many ways, compared to Trump’s various definitions, with little overlap between the paper and the former president. Editorially, the paper emphasizes conventional definitions of fake news that avoid understanding it as a form of propaganda operating in service of governmental interests. I examine various competing definitions of fake news in other media venues, providing evidence that the social construction of fake news is a contested phenomenon. I examine US partisan cable media, alternative left- and right-wing media, and social media venues – each of which puts forward its own interpretations of what fake news means.

Deligne cohomology is a tool that makes it possible to unify the study of cycles through an object that classifies extensions of (p, p)-cycles by points in the p-th intermediate Jacobian (which is the target of the Abel–Jacobi map on cycles of codimension p). This is treated in Section 10.1 with applications to normal functions.

Before giving the proof of Nori's theorem in Section 10.6, we need some results from mixed Hodge theory. These are proven in Section 10.2 where we also state different variants of the theorem. Sections 10.3 and 10.4 treat a localto- global principle and an extension of the method of Jacobian representations of cohomology which are both essential for the proof. We finish the chapter with some applications of Nori's theorem and discuss the conjectured filtrations on the Chow groups to which these lead.

A Detour into Deligne Cohomology with Applications

Here we introduce Deligne cohomology in the form first defined by P. Deligne. We illustrate its connections to intermediate Jacobians and explain its functorial properties. Then we give the second proof of Theorem 9.1.3. The results of this section are not needed for an understanding of the rest of the chapter, and some readers may want to skip this section and read it later. In the historical remarks at the end of the chapter we point out some further directions where Deligne–Beilinson cohomology becomes more important. Deligne cohomology was defined first by P. Deligne and later extended by A. Beilinson. We use here mainly the original version of Deligne without growth conditions for noncompact spaces. Beilinson later imposed such growth conditions in order to get a more functorial theory. The extension of Beilinson has been worked out in detail in Esnault and Viehweg (1988).

Definition 10.1.1 Let X be a Kahler manifold. Define the analytic Deligne complex on X by

This is a complex of sheaves in the analytic topology and we put the first sheaf (2πi)pZ, which is a constant subsheaf of C, in degree 0. Hence the last sheaf, sits in degree p. We denote by

the 2p-th hypercohomology of this complex and call it the Deligne cohomology group of X.

In this chapter we treat Mumford–Tate domains in an axiomatic fashion. Historically, the concept of “Shimura domain” arose first. Such domains have a representation as bounded symmetric domains. The tautological variation of Hodge flag over such a domain satisfies Griffiths’ transversality, i.e.,we have a variation of Hodge structure over it. This characterizes them among Mumford–Tate domains and makes them easier to study directly from a Hodge theoretic perspective, which we do in Section 16.1. To make the transition to the axiomatic treatment, one views a Mumford–Tate domain as an entire conjugacy class of a given Hodge structure. To get a polarizable Hodge structure, the connected group of automorphisms of the domain must be a reductive group of Hodge type and, conversely, these are the groups that act transitively on Mumford–Tate domains. We explain this in Section 16.2. In Section 16.3 we give the promised axiomatic treatment of Mumford–Tate varieties parallel to Deligne's axiomatic treatment of Shimura varieties. The chapter ends with Section 16.4 where we give examples of Hodge structures given by representations of the classical simple groups.

Shimura Domains

Basic Properties and Classification

Definition 16.1.1 A Shimura domain is a Mumford–Tate subdomain D of some period domain such that the tautological Hodge flag restricts to a variation of Hodge structure on D.

Remark It may very well happen that the same Shimura domain can be embedded in different period domains and so the tautological variation can a priori be different. However, there is a way to give a more abstract treatment of Shimura domains which starts with the algebraic group under which the domain is homogeneous. See Proposition 16.3.3.

Our main result concerning Shimura domains is Proposition 16.1.9. Before we state it, let us first give some examples.

In the fourteen years since the first edition appeared, ample experience with teaching to graduate students made us realize that a proper understanding of several of the core aspects of period domains needed a lot more explanation than offered in the first edition of this book, especially with regards to the Lie group aspects of period domains.

Consequently, we decided a thorough reworking of the book was called for. In particular Section 4.3, and Chapters 12 and 13 needed revision. The latter two chapters have been rearranged and now contain more, often completely rewritten sections. At the same time relevant newer developments have been inserted at appropriate places. Finally we added a new “Part Four” with additional, more recent topics. This also required an extra Appendix D about Lie groups and algebraic groups.

Let us be more specific about the added material. There is a new Section 5.4 on counterexamples to infinitesimal Torelli. In Chapter 6 the abstract and powerful formalism of derived functors has been added so that for instance the algebraic treatment of the Gauss–Manin connection could be given, as well as a proper treatment of the Leray spectral sequence. In Chapter 13 we have devoted more detail on Higgs bundles and their logarithmic variant. This made it possible to also include some geometric applications at the end of that chapter.

“Part Four” starts with a chapter explaining the by now standard group theoretic formulation of the concept of a Hodge structure. This naturally leads to Mumford–Tate groups and their associated domains. The chapter culminates with a result giving a factorization of the period map which stresses the role of the Mumford–Tate group of a given variation. In Chapter 16 Mumford– Tate domains and their quotients by certain discrete groups, the Mumford–Tate varieties, are considered from a more abstract, axiomatic point of view. In this chapter the relation with the classical Shimura varieties is also explained. In the next and final chapter we study various interesting subvarieties of Mumford– Tate varieties, especially of low dimension.

One word about the prerequisites. Of course, they remain the same (see page xi), but we should mention a couple of more recent books that may serve as a guide to the reader.