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Periods and Nori Motives 1st ed. 2017 [Kietas viršelis]

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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 perio

d 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. 

Part I Background Material.- General Set-Up.- Singular Cohomology.- Algebraic de Rham Cohomology.- Holomorphic de Rham Cohomology.- The Period Isomorphism.- Categories of (Mixed) Motives.- Part II Nori Motives.- Nori"s Diagram Category.- More on Diagrams.- Nori Motives.- Weights and Pure Nori Motives.- Part III Periods.- Periods of Varieties.- Kontsevich-Zagier Periods.- Formal Periods and the Period Conjecture.- Part IV Examples.- Elementary Examples.- Multiple Zeta Values.- Miscellaneous Periods: an Outlook.

Recenzijos

This book is admirably suited for guiding a course or seminar program on this topic. The authors are to be congratulated on producing an important contribution to the mathematical literature on motives and their application to central problems in algebraic geometry and arithmetic. (Marc Levine, Jahresbericht der Deutschen Mathematiker-Vereinigung, August 13, 2019) The book under review provides a detailed account on some of the theory of so-called Nori motives . The authors provide a lot of details and background information, making this book very accessible. this book is a valuable contribution to the field of motives. Particularly commendable is the attention to detail, which can sometimes be missing in this field riddled with conjectures and folklore results. The expository nature makes this book useful to a wide audience. (Tom Bachmann, zbMATH 1369.14001, 2017)This text is both a stimulating introduction and a sound comprehensive reference for anyone interested in the field of motives and periods. All things considered, I strongly feel that the authors deserve praise for their valiant work. They have fulfilled their difficult program bravely and efficiently. (Alberto Collino, Mathematical Reviews, 2017)

Part I Background Material
1 General Set-Up
3(28)
1.1 Varieties
3(2)
1.1.1 Linearising the Category of Varieties
3(1)
1.1.2 Divisors with Normal Crossings
4(1)
1.2 Complex Analytic Spaces
5(1)
1.2.1 Analytification
5(1)
1.3 Complexes
6(3)
1.3.1 Basic Definitions
6(1)
1.3.2 Filtrations
7(1)
1.3.3 Total Complexes and Signs
8(1)
1.4 Hypercohomology
9(6)
1.4.1 Definition
10(1)
1.4.2 Godement Resolutions
11(2)
1.4.3 Cech Cohomology
13(2)
1.5 Simplicial Objects
15(5)
1.6 Grothendieck Topologies
20(2)
1.7 Torsors
22(9)
1.7.1 Sheaf-Theoretic Definition
23(1)
1.7.2 Torsors in the Category of Sets
24(3)
1.7.3 Torsors in the Category of Schemes (Without Groups)
27(4)
2 Singular Cohomology
31(42)
2.1 Relative Cohomology
31(3)
2.2 Singular (Co)homology
34(2)
2.3 Simplicial Cohomology
36(5)
2.4 The Kunneth Formula and Poincare Duality
41(4)
2.5 The Basic Lemma
45(14)
2.5.1 Formulations of the Basic Lemma
45(2)
2.5.2 Direct Proof of Basic Lemma I
47(2)
2.5.3 Nori's Proof of Basic Lemma II
49(3)
2.5.4 Beilinson's Proof of Basic Lemma II
52(3)
2.5.5 Perverse Sheaves and Artin Vanishing
55(4)
2.6 Triangulation of Algebraic Varieties
59(11)
2.6.1 Semi-algebraic Sets
60(6)
2.6.2 Semi-algebraic Singular Chains
66(4)
2.7 Singular Cohomology via the h'-Topology
70(3)
3 Algebraic de Rham Cohomology
73(24)
3.1 The Smooth Case
73(10)
3.1.1 Definition
73(3)
3.1.2 Functoriality
76(1)
3.1.3 Cup Product
77(2)
3.1.4 Change of Base Field
79(1)
3.1.5 Etale Topology
80(1)
3.1.6 Differentials with Log Poles
81(2)
3.2 The General Case: Via the h-Topology
83(4)
3.3 The General Case: Alternative Approaches
87(10)
3.3.1 Deligne's Method
87(3)
3.3.2 Hartshorne's Method
90(1)
3.3.3 Using Geometric Motives
91(3)
3.3.4 The Case of Divisors with Normal Crossings
94(3)
4 Holomorphic de Rham Cohomology
97(10)
4.1 Holomorphic de Rham Cohomology
97(5)
4.1.1 Definition
97(2)
4.1.2 Holomorphic Differentials with Log Poles
99(1)
4.1.3 GAGA
100(2)
4.2 Holomorphic de Rham Cohomology via the h'-Topology
102(5)
4.2.1 H'-Differentials
102(1)
4.2.2 Holomorphic de Rham Cohomology
103(1)
4.2.3 GAGA
104(3)
5 The Period Isomorphism
107(10)
5.1 The Category (k, Q)--Vect
107(1)
5.2 A Triangulated Category
108(1)
5.3 The Period Isomorphism in the Smooth Case
109(2)
5.4 The General Case (via the h'-Topology)
111(2)
5.5 The General Case (Deligne's Method)
113(4)
6 Categories of (Mixed) Motives
117(20)
6.1 Pure Motives
117(2)
6.2 Geometric Motives
119(5)
6.3 Absolute Hodge Motives
124(5)
6.4 Mixed Tate Motives
129(8)
Part II Nori Motives
7 Nori's Diagram Category
137(40)
7.1 Main Results
137(8)
7.1.1 Diagrams and Representations
137(2)
7.1.2 Explicit Construction of the Diagram Category
139(1)
7.1.3 Universal Property: Statement
140(4)
7.1.4 Discussion of the Tannakian Case
144(1)
7.2 First Properties of the Diagram Category
145(4)
7.3 The Diagram Category of an Abelian Category
149(16)
7.3.1 A Calculus of Tensors
150(6)
7.3.2 Construction of the Equivalence
156(8)
7.3.3 Examples and Applications
164(1)
7.4 Universal Property of the Diagram Category
165(3)
7.5 The Diagram Category as a Category of Comodules
168(9)
7.5.1 Preliminary Discussion
168(1)
7.5.2 Coalgebras and Comodules
169(8)
8 More on Diagrams
177(30)
8.1 Multiplicative Structure
177(11)
8.2 Localisation
188(3)
8.3 Nori's Rigidity Criterion
191(4)
8.4 Comparing Fibre Functors
195(12)
8.4.1 The Space of Comparison Maps
196(5)
8.4.2 Some Examples
201(3)
8.4.3 The Description as Formal Periods
204(3)
9 Nori Motives
207(26)
9.1 Essentials of Nori Motives
207(5)
9.1.1 Definition
207(2)
9.1.2 Main Results
209(3)
9.2 Yoga of Good Pairs
212(8)
9.2.1 Good Pairs and Good Filtrations
212(1)
9.2.2 Tech Complexes
213(3)
9.2.3 Putting Things Together
216(2)
9.2.4 Comparing Diagram Categories
218(2)
9.3 Tensor Structure
220(6)
9.3.1 Collection of Proofs
225(1)
9.4 Artin Motives
226(2)
9.5 Change of Fields
228(5)
10 Weights and Pure Nori Motives
233(14)
10.1 Comparison Functors
233(3)
10.2 Weights and Nori Motives
236(5)
10.2.1 Andre's Motives
237(1)
10.2.2 Weights
238(3)
10.3 Tate Motives
241(6)
Part III Periods
11 Periods of Varieties
247(14)
11.1 First Definition
247(3)
11.2 Periods for the Category (k, Q)--Vect
250(3)
11.3 Periods of Algebraic Varieties
253(3)
11.3.1 Definition
253(2)
11.3.2 First Properties
255(1)
11.4 The Comparison Theorem
256(2)
11.5 Periods of Motives
258(3)
12 Kontsevich-Zagier Periods
261(12)
12.1 Definition
261(4)
12.2 Comparison of Definitions of Periods
265(8)
13 Formal Periods and the Period Conjecture
273(16)
13.1 Formal Periods and Nori Motives
273(4)
13.2 The Period Conjecture
277(10)
13.2.1 Formulation in the Number Field Case
278(1)
13.2.2 Consequences
279(3)
13.2.3 Special Cases and the Older Literature
282(2)
13.2.4 The Function Field Case
284(3)
13.3 The Case of 0-Dimensional Varieties
287(2)
Part IV Examples
289(66)
14 Elementary Examples
291(16)
14.1 Logarithms
291(2)
14.2 More Logarithms
293(1)
14.3 Quadratic Forms
294(3)
14.4 Elliptic Curves
297(4)
14.5 Periods of 1-Forms on Arbitrary Curves
301(6)
15 Multiple Zeta Values
307(30)
15.1 A ζ-value, the Basic Example
307(3)
15.2 Definition of Multiple Zeta Values
310(2)
15.3 Kontsevich's Integral Representation
312(2)
15.4 Relations Among Multiple Zeta Values
314(6)
15.5 Multiple Zeta Values and Moduli Space of Marked Curves
320(1)
15.6 Multiple Polylogarithms
321(16)
15.6.1 The Configuration
322(1)
15.6.2 Singular Homology
323(3)
15.6.3 Smooth Singular Homology
326(1)
15.6.4 Algebraic de Rham Cohomology and the Period Matrix of (X, D)
327(4)
15.6.5 Varying the Parameters a and b
331(6)
16 Miscellaneous Periods: An Outlook
337(18)
16.1 Special Values of L-Functions
337(4)
16.2 Feynman Periods
341(2)
16.3 Algebraic Cycles and Periods
343(4)
16.4 Periods of Homotopy Groups
347(2)
16.5 Exponential Periods
349(1)
16.6 Non-periods
350(5)
Glossary 355(4)
References 359(10)
Index 369
Annette Huber works in arithmetic geometry, in particular on motives and special values of L-functions. She has contributed to all aspects of the Bloch-Kato conjecture, a vast generalization of the class number formula and the conjecture of Birch and Swinnerton-Dyer. More recent research interests include period numbers in general and differential forms on singular varieties.

Stefan Müller-Stach works in algebraic geometry, focussing on algebraic cycles, regulators and period integrals. His work includes the detection of classes in motivic cohomology via regulators and the study of special subvarieties in Mumford-Tate varieties. More recent research interests include periods and their relations to mathematical physics and foundations of mathematics.