http://www.loc.gov/mods/v3 http://www.loc.gov/standards/mods/v3/mods-3-2.xsd
1 online resource (xxiii, 372 pages) : illustrations
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.
Preface, with an Extended Introduction; Part I Background Material; 1 General Set-Up; 1.1 Varieties; 1.1.1 Linearising the Category of Varieties; 1.1.2 Divisors with Normal Crossings; 1.2 Complex Analytic Spaces; 1.2.1 Analytification; 1.3 Complexes; 1.3.1 Basic Definitions; 1.3.2 Filtrations; 1.3.3 Total Complexes and Signs; 1.4 Hypercohomology; 1.4.1 Definition; 1.4.2 Godement Resolutions; 1.4.3 Čech Cohomology; 1.5 Simplicial Objects; 1.6 Grothendieck Topologies; 1.7 Torsors; 1.7.1 Sheaf-Theoretic Definition; 1.7.2 Torsors in the Category of Sets.
1.7.3 Torsors in the Category of Schemes (Without Groups)2 Singular Cohomology; 2.1 Relative Cohomology; 2.2 Singular (Co)homology; 2.3 Simplicial Cohomology; 2.4 The Künneth Formula and Poincaré Duality; 2.5 The Basic Lemma; 2.5.1 Formulations of the Basic Lemma; 2.5.2 Direct Proof of Basic Lemma I; 2.5.3 Nori's Proof of Basic Lemma II; 2.5.4 Beilinson's Proof of Basic Lemma II; 2.5.5 Perverse Sheaves and Artin Vanishing; 2.6 Triangulation of Algebraic Varieties; 2.6.1 Semi-algebraic Sets; 2.6.2 Semi-algebraic Singular Chains; 2.7 Singular Cohomology via the h'-Topology.
3 Algebraic de Rham Cohomology3.1 The Smooth Case; 3.1.1 Definition; 3.1.2 Functoriality; 3.1.3 Cup Product; 3.1.4 Change of Base Field; 3.1.5 Étale Topology; 3.1.6 Differentials with Log Poles; 3.2 The General Case: Via the h-Topology; 3.3 The General Case: Alternative Approaches; 3.3.1 Deligne's Method; 3.3.2 Hartshorne's Method; 3.3.3 Using Geometric Motives; 3.3.4 The Case of Divisors with Normal Crossings; 4 Holomorphic de Rham Cohomology; 4.1 Holomorphic de Rham Cohomology; 4.1.1 Definition; 4.1.2 Holomorphic Differentials with Log Poles; 4.1.3 GAGA.
4.2 Holomorphic de Rham Cohomology via the h'-Topology4.2.1 h'-Differentials; 4.2.2 Holomorphic de Rham Cohomology; 4.2.3 GAGA; 5 The Period Isomorphism; 5.1 The Category (k, mathbbQ)-Vect; 5.2 A Triangulated Category; 5.3 The Period Isomorphism in the Smooth Case; 5.4 The General Case (via the h'-Topology); 5.5 The General Case (Deligne's Method); 6 Categories of (Mixed) Motives; 6.1 Pure Motives; 6.2 Geometric Motives; 6.3 Absolute Hodge Motives; 6.4 Mixed Tate Motives; Part II Nori Motives; 7 Nori's Diagram Category; 7.1 Main Results; 7.1.1 Diagrams and Representations.
7.1.2 Explicit Construction of the Diagram Category7.1.3 Universal Property: Statement; 7.1.4 Discussion of the Tannakian Case; 7.2 First Properties of the Diagram Category; 7.3 The Diagram Category of an Abelian Category; 7.3.1 A Calculus of Tensors; 7.3.2 Construction of the Equivalence; 7.3.3 Examples and Applications; 7.4 Universal Property of the Diagram Category; 7.5 The Diagram Category as a Category of Comodules; 7.5.1 Preliminary Discussion; 7.5.2 Coalgebras and Comodules; 8 More on Diagrams; 8.1 Multiplicative Structure; 8.2 Localisation; 8.3 Nori's Rigidity Criterion.
Annette Huber, Stefan Müller-Stach ; with contributions by Benjamin Friedrich and Jonas von Wangenheim.
Includes bibliographical references and index.
Ergebnisse der Mathematik und ihrer Grenzgebiete ; 3. Folge, Bd. 65
com.springer.onix.9783319509266 Springer Nature