Periods and Nori Motives (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics Book 65)

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. "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." -- Provided by publisher Front Matter....Pages i-xxiii Front Matter....Pages 1-1 General Set-Up....Pages 3-29 Singular Cohomology....Pages 31-72 Algebraic de Rham Cohomology....Pages 73-96 Holomorphic de Rham Cohomology....Pages 97-105 The Period Isomorphism....Pages 107-116 Categories of (Mixed) Motives....Pages 117-133 Front Matter....Pages 135-135 Nori’s Diagram Category....Pages 137-175 More on Diagrams....Pages 177-206 Nori Motives....Pages 207-232 Weights and Pure Nori Motives....Pages 233-243 Front Matter....Pages 245-245 Periods of Varieties....Pages 247-259 Kontsevich–Zagier Periods....Pages 261-272 Formal Periods and the Period Conjecture....Pages 273-288 Front Matter....Pages 289-289 Elementary Examples....Pages 291-305 Multiple Zeta Values....Pages 307-336 Miscellaneous Periods: An Outlook....Pages 337-353 Back Matter....Pages 355-372