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Periods and Nori motives / Annette Huber, Stefan Müller-Stach ; with contributions by Benjamin Friedrich and Jonas von Wangenheim. - online (ELvl I)
Cham, Switzerland : Springer Nature, 2017.
xxiii, 372 pages : illustrations ; 24 cm.
Müller-Stach, Stefan, 1962-
Ergebnisse der Mathematik und ihrer Grenzgebiete ; 3. Folge, Bd. 65.
[More in this series]
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, 0071-1136 ; volume 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."--Provided by publisher.
Includes bibliographical references (pages 359-367) and index.
General set-up -- Singular cohomology -- Algebraic de Rahm cohomology -- The period isomorphism -- Categories of (mixed) motives -- Nori's diagram category -- More on diagrams -- Nori motives -- Weights and pure Nori motives -- Periods of varieties -- Kontsevich-Zagier periods -- Formal periods and the period conjecture -- Elementary examples -- Multiple zeta values -- Miscellaneous periods : an outlook.
Also available in an electronic version.
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Periods and Nori Motives [electronic resource] / by Annette Huber, Stefan Müller-Stach.