Periods and Nori Motives

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This book explores the theory of periods of algebraic varieties within Madhav Nori’s abelian category of mixed motives. It develops Nori’s approach from the ground up, addressing a significant gap in existing literature, and elaborates on the relationship between mixed motives and periods, including a comprehensive discussion of period numbers as defined by Kontsevich-Zagier and their structural characteristics. Period numbers are crucial in number theory and algebraic geometry and also have implications in mathematical physics. Long-standing conjectures regarding their transcendence properties are best understood through the cohomology of algebraic varieties or motives. The book highlights Nori’s construction of an abelian category of motives over fields that can be embedded into the complex numbers, which is particularly effective for this analysis. Additionally, it shows how Kontsevich's formal period algebra functions as a torsor under the motivic Galois group in Nori's framework, allowing for a rephrasing of the Kontsevich-Zagier period conjecture. This informative text is aimed at graduate students in algebraic geometry and number theory, as well as researchers in related disciplines. It includes essential background on singular cohomology, algebraic de Rham cohomology, diagram categories, and rigid tensor categories, along with numerous examples, making it a self-contained resource.