Periods and Nori Motives

Annette Huber, Benjamin Friedrich, Stefan Müller-Stach , Jonas von Wangenheim

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English

Springer International Publishing AG
21 July 2018

Algebra; Number theory; Algebraic geometry; Algebraic topology

Series: Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics

This book casts the theory of periods of algebraic varieties in the natural setting of Madhav Nori’s abelian category of mixed motives. It develops Nori’s approach to mixed motives from scratch, thereby filling an important gap in the literature, and then explains the connection of mixed motives to periods, including a detailed account of the theory of period numbers in the sense of Kontsevich-Zagier and their structural properties. Period numbers are central to number theory and algebraic geometry, and also play an important role in other fields such as mathematical physics. There are long-standing conjectures about their transcendence properties, best understood in the language of cohomology of algebraic varieties or, more generally, motives. Readers of this book will discover that Nori’s unconditional construction of an abelian category of motives (over fields embeddable into the complex numbers) is particularly well suited for this purpose. Notably, Kontsevich's formal period algebra represents a torsor under the motivic Galois group in Nori's sense, and the period conjecture of Kontsevich and Zagier can be recast in this setting. Periods and Nori Motives is highly informative and will appeal to graduate students interested in algebraic geometry and number theory as well as researchers working in related fields. Containing relevant background material on topics such as singular cohomology, algebraic de Rham cohomology, diagram categories and rigid tensor categories, as well as many interesting examples, the overall presentation of this book is self-contained.

By: Annette Huber, Stefan Müller-Stach
Contributions by: Benjamin Friedrich, Jonas von Wangenheim
Imprint: Springer International Publishing AG
Country of Publication: Switzerland
Edition: Softcover reprint of the original 1st ed. 2017
Volume: 65
Dimensions: Height: 235mm, Width: 155mm,
Weight: 5.971kg
ISBN: 9783319845241
ISBN 10: 3319845241
Series: Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics
Pages: 372
Publication Date: 21 July 2018
Audience: Professional and scholarly , Undergraduate
Format: Paperback
Publisher's Status: Active

Part I Background Material.- General Set-Up.- Singular Cohomology.- Algebraic de Rham Cohomology.- Holomorphic de Rham Cohomology.- The Period Isomorphism.- Categories of (Mixed) Motives.- Part II Nori Motives.- Nori's Diagram Category.- More on Diagrams.- Nori Motives.- Weights and Pure Nori Motives.- Part III Periods.- Periods of Varieties.- Kontsevich–Zagier Periods.- Formal Periods and the Period Conjecture.- Part IV Examples.- Elementary Examples.- Multiple Zeta Values.- Miscellaneous Periods: an Outlook.

Annette Huber works in arithmetic geometry, in particular on motives and special values of L-functions. She has contributed to all aspects of the Bloch-Kato conjecture, a vast generalization of the class number formula and the conjecture of Birch and Swinnerton-Dyer. More recent research interests include period numbers in general and differential forms on singular varieties. Stefan Müller-Stach works in algebraic geometry, focussing on algebraic cycles, regulators and period integrals. His work includes the detection of classes in motivic cohomology via regulators and the study of special subvarieties in Mumford-Tate varieties. More recent research interests include periods and their relations to mathematical physics and foundations of mathematics.

Reviews for Periods and Nori Motives

The book under review provides a detailed account on some of the theory of so-called Nori motives ... . The authors provide a lot of details and background information, making this book very accessible. ... 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. (Tom Bachmann, zbMATH 1369.14001, 2017) This text is both a stimulating introduction and a sound comprehensive reference for anyone interested in the field of motives and periods. ... All things considered, I strongly feel that the authors deserve praise for their valiant work. They have fulfilled their difficult program bravely and efficiently. (Alberto Collino, Mathematical Reviews, 2017)