General Set-Up

The Period Isomorphism

The aim of this section is to define well-behaved isomorphisms between singular and de Rham cohomology of algebraic varieties.

Algebraic de Rham Cohomology

Let k be a field of characteristic zero. We are going to define relative algebraic de Rham cohomology for general varieties over k, not necessarily smooth.

Holomorphic de Rham Cohomology

We are going to define a natural comparison isomorphism between algebraic de Rham cohomology and singular cohomology of varieties over the complex numbers with coefficients in \(\mathbb {C}\). The link is provided by holomorphic de Rham cohomology, which we study in this chapter.

Miscellaneous Periods: An Outlook

In this chapter, we collect several other important examples of periods in the literature for the convenience of the reader.

Periods of Varieties

A period, or more precisely, a period number may be thought of as the value of an integral that occurs in a geometric context. In their papers [Kon99, KZ01], Kontsevich and Zagier list various ways of defining a period. We show that all these variants give the same notion.

Weights and Pure Nori Motives

In this chapter, we explain how Nori motives relate to other categories of motives. By the work of Harrer, the realisation functor from geometric motives to absolute Hodge motives factors via Nori motives. We then use this in order to establish the existence of a weight filtration on Nori motives with rational coefficients. The category of pure Nori motives turns out to be equivalent to Andre’s category of motives via motivated cycles.

More on Diagrams

The aim of this chapter is to introduce and study additional structures on a diagram such that its diagram category becomes a rigid tensor category. The assumptions are tailored to the application to Nori motives.

Kontsevich–Zagier Periods

We compare the set of Kontsevich–Zagier periods defined by integrals over semi-algebraic subsets of \(\mathbb {R}^n\) with cohomological periods.

Nori’s Diagram Category

We explain Nori’s construction of an abelian category attached to the representation of a diagram and establish some properties for it. The construction is completely formal. It mimics the standard construction of the Tannakian dual of a rigid tensor category with a fibre functor . Only, we do not have a tensor product or even a category but only what we should think of as the fibre functor.

Formal Periods and the Period Conjecture

Following Kontsevich (see Kontsevich in Operads and motives in deformation quantization. Lett. Math. Phys. 48(1):35–72, 1999), we now introduce another algebra \(\tilde{\mathbb {P}}(k)\) of formal periods from the same data we have used in order to define the actual period algebra of a field in Chap. 11. The main aim of this chapter is to give conceptual interpretation of this algebra of formal periods. We then use it to formulate and discuss the period conjecture.

Multiple Zeta Values

We study in some detail the very important class of periods called multiple zeta values (MZV). These are periods of mixed Tate motives, which we discussed in Sect. 6.4. Multiple zeta values are in fact periods of unramified mixed Tate motives, a full subcategory of all mixed Tate motives.

Categories of (Mixed) Motives