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.
