The book under review provides a detailed account on some of the theory of so-called Nori motives, in particular as to how they relate to periods. The authors provide a lot of details and background information, making this book very accessible.
The book consists of four parts. The first part gathers background material: standard results like hypercohomology of sheaves, Grothendieck topologies, algebraic de Rham cohomology, GAGA, and so on. There are also many less standard topics treated: the h-topology on schemes and its connection to the algebraic de Rham cohomology of singular varieties, simplicial objects, etc. There are also results specific to the subject matter of motives, such as Nori’s basic lemma, and a quick introduction to various constructions of categories of motives.
Part II sets up the general theory of Nori motives. The idea can be summarized as follows. The conjectural category \(MM^{eff}(k)\) of effective mixed motives over a field \(k \subset \mathbb{C}\) should be an abelian category together with contravariant functors \(H^i: Var(k)^{op} \to MM^{eff}(k)\), satisfying certain properties. Here \(Var(k)\) is the category of varieties over \(k\) (not necessariliy smooth or projective). If \(X \in Var(k)\), \(H^i(X)\) is called the motive of \(X\) in dimension \(i\). More precisely, for every pair of varieties \((X, Z)\) with \(Z \subset X\) closed, there should be a relative motive \(H^i(X, Z) \in MM^{eff}(k)\). Among the axioms should be the following: (1) if \(f: X' \to X\) is a morphism of varieties and \(Z' \subset X', Z \subset X\) with \(f(Z') \subset f(Z)\), then there is an induced morphism \(f^*: H^i(X, Z) \to H^i(X', Z')\), (2) if \(Z \subset Y \subset X\), then there is a morphism \(\partial: H^i(Y, Z) \to H^{i+1}(X, Y)\), and (3) the functor \(H^*: Var(k)^{op} \to Ab\) of (relative) singular cohomology factors faithfully through \(H^*: Var(k)^{op} \to MM^{eff}(k)\). In other words there exists a functor \(H^*: MM^{eff}(k) \to Ab\) which sends \(H^i(X, Z) \in MM^{eff}(k)\) to the relative singular cohomology group of dimension \(i\) of the pair \((X(\mathbb{C}), Z(\mathbb{C}))\), and this induced functor is faithful and exact.
Let us note that quite a lot can be concluded from these axioms. For example, faithful exact functors are conservative. Consequently, the maps \(\partial\) from axiom (2) fit into long exact sequences, since the same holds true for ordinary singular cohomology.
It is expected that the category \(MM^{eff}(k)\) has a symmetric monoidal structure \(\otimes\) coming from the cartesian product of varieties. Then the full category of mixed motives \(MM(k)\) is supposed to be obtained by inverting the Lefschetz motive \(L = H^1(\mathbb{P}^1)\) under \(\otimes\).
Nori’s insight was that there is actually a universal abelian category satisfying properties (1) to (3), and that this category can be constructed explicitly and studied in some detail. This is what is carried out in part II. In particular it is shown that there is a symmetric monoidal structure, and that inverting \(L\) yields a rigid tensor category. Also many formal properties of this kind of construction are established, including generalisations of the Tannakian reconstruction formalism. This in particular constructs motivic fundamental groups.
Part III introduces the second main topic, periods. Periods come in two rather different flavours. The first flavour, the set periods of varieties over \(k\), is just a certain subring of \(\mathbb{C}\); it essentially consists of the values obtained by integrating algebraic differential forms over algebraic chains on algebraic varieties. All algebraic numbers are periods in this sense, as are many interesting transcendental numbers, like \(\pi\) or \(\zeta(2)\). The set of periods is denoted \(\mathbb{P}(k) \subset \mathbb{C}\). The authors show that various possible definitions of \(\mathbb{P}(k)\) coincide, and establish basic properties, such as that the set is a ring. The second flavour is the so-called formal periods \(\tilde\mathbb{P}(k)\). This is essentially the ring generated by triples \((X, \alpha, \omega)\) where \(X\) is an algebraic variety, \(\alpha\) is an algebraic chain, and \(\omega\) is an algebraic differential form, modulo certain relations. By construction, integrating \(\omega\) against \(\alpha\) yields a homomorphism \(\tilde\mathbb{P}(k) \to \mathbb{P}(k)\). The period conjecture states that this map is an isomorphism. The authors provide a detailed proof of the fact that \(\tilde\mathbb{P}(k)\) can be made into an algebraic variety over \(k\), which is closely related to the motivic fundamental group of \(k\).
Chapter IV treats many examples of periods, such as those coming from abelian varieties, polyologarithms, and multiple zeta values, all using the methods that have been carefully established in the previous parts of the book.
Overall, this book is a valuable contribution to the field of motives. Particularly commendable is the attention to detail, which can sometimes be missing in this field riddled with conjectures and folklore results. The expository nature makes this book useful to a wide audience.