Periods and Nori Motives / by Annette Huber, Stefan Müller-Stach.

Format:

Book

Author/Creator:

Huber, Annette, Author.

Müller-Stach, Stefan., Author.

Series:

Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, 0071-1136 ; 65

Language:

English

Subjects (All):

Number theory.

Geometry, Algebraic.

K-theory.

Algebraic topology.

Categories (Mathematics).

Algebra, Homological.

Associative rings.

Rings (Algebra).

Number Theory.

Algebraic Geometry.

K-Theory.

Algebraic Topology.

Category Theory, Homological Algebra.

Associative Rings and Algebras.

Local Subjects:

Number Theory.

Algebraic Geometry.

K-Theory.

Algebraic Topology.

Category Theory, Homological Algebra.

Associative Rings and Algebras.

Physical Description:

1 online resource (XXIII, 372 p. 7 illus.)

Edition:

1st ed. 2017.

Place of Publication:

Cham : Springer International Publishing : Imprint: Springer, 2017.

Summary:

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.

Contents:

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.

Notes:

Includes bibliographical references and index.