Periods and Nori motives
Annette Huber and Stefan Müller-Stach; with contributions of
Benjamin Friedrich and Jonas von Wangenheim
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 65. Springer, Cham, 2017.
In this book explain the construction of
Nori's category of motives, its relation to the period algebras as well
as the necessary background material like singular and de Rham cohomology.
It is based on the preprint "On the relation between Nori Motives and Kontsevich Periods'' (Huber, Müller-Stach) and the diploma theses
"Nori-Motive und Tannaka-Theorie" (von Wangenheim) and "Periods and algebraic deRham cohomology'' (Friedrich). All are available on arXiv.
Here is an annotated version as of March, 2, 2022 (pdf) containing corrections of the mistakes we are aware of.
The text in black is identical to the printed book, but the typesetting, in particular, page numbers, are not. New additions/errata are in blue.
We thank the many people who sent us comments and corrections also on earlier versions.
List of Errata
- (communicated by D. Cardaris) The formula for the relation between multiple zeta values in Example 15.4.4 contains a wrong factor.
- (communicated by D.-C. Cisinski) Statement and Proof of Prop. 6.2.14 on Bondardko's weight structures garbled; only a weaker statement holds. Misprint in Theorem 6.2.1, corrected. As a consequence, Corollary 10.2.6 also has to weakened. The application to Theorem 10.2.7 (characterisation of pure Nori motives) remains valid.
- The proof Thm 11.4.2 has a mistake. The statement is correct, argument sketched.
- (communicated by Y. André) Thm 9.1.6. Proof incomplete, now fixed.
- (communicated by Kapil Paranjape) Chapter 8 needs to assume that we are working over a Dedekind domain.
- (communicated by Y. André) Prop. 10.2.1, only a weaker statement is proved.
- Prop. 13.1.11, proof incomplete, now fixed.
- (communicated by F. Brown) Formula (14.7), statement wrong
Part I Background material
1 General Set-up
2 Singular cohomology
3 Algebraic de Rham cohomology
4 Holomorphic de Rham cohomology
5 The period isomorphism
6 Categories of (mixed) motives
Part II Nori Motives
7 Nori's digram category
8 More on diagrams
9 Nori motives
10 Weights and pure Nori motives
Part III Periods
11 Periods of varieties
12 Kontsevich-Zagier Periods
13 Formal Periods and the period conjecture
Part IV Examples
14 Elementary examples
15 Multiple zeta-values
16 Miscellaneous periods: an outlook