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Motivic Integration Softcover Reprint of the Original 1st 2018 ed. [Minkštas viršelis]

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  • Formatas: Paperback / softback, 526 pages, aukštis x plotis: 235x155 mm, weight: 831 g, 47 Illustrations, black and white; XX, 526 p. 47 illus., 1 Paperback / softback
  • Serija: Progress in Mathematics 325
  • Išleidimo metai: 10-Dec-2019
  • Leidėjas: Birkhauser Boston Inc
  • ISBN-10: 1493993151
  • ISBN-13: 9781493993154
Kitos knygos pagal šią temą:
Motivic Integration Softcover Reprint of the Original 1st 2018 ed.Didesnis paveikslėlis
  • Formatas: Paperback / softback, 526 pages, aukštis x plotis: 235x155 mm, weight: 831 g, 47 Illustrations, black and white; XX, 526 p. 47 illus., 1 Paperback / softback
  • Serija: Progress in Mathematics 325
  • Išleidimo metai: 10-Dec-2019
  • Leidėjas: Birkhauser Boston Inc
  • ISBN-10: 1493993151
  • ISBN-13: 9781493993154
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9781493993154.html
This monograph focuses on the geometric theory of motivic integration, which takes its values in the Grothendieck ring of varieties. This theory is rooted in a groundbreaking idea of Kontsevich and was further developed by Denef & Loeser and Sebag. It is presented in the context of formal schemes over a discrete valuation ring, without any restriction on the residue characteristic. The text first discusses the main features of the Grothendieck ring of varieties, arc schemes, and Greenberg schemes. It then moves on to motivic integration and its applications to birational geometry and non-Archimedean geometry. Also included in the work is a prologue on p-adic analytic manifolds, which served as a model for motivic integration. 

With its extensive discussion of preliminaries and applications, this book is an ideal resource for graduate students of algebraic geometry and researchers of motivic integration. It will also serve as a motivation for more recent and sophisticated theories that have been developed since. 

Recenzijos

The introduction contains a very succinct presentation of the contents of each chapter, which efficiently gives a first rough idea of how everything is presented and organised. , the author of this review believes that a more detailed account of some of the chapters, somehow designed as a mathematical tour guide, could be useful to others, keeping it as much readable as possible and focusing on the main concepts introduced in this very rich book. (Loļs Faisant, zbMATH 1545.14021, 2025)

1 Prologue: p-Adic Integration
1(54)
§ 1 Analytic Manifolds
1(24)
1.1 Local Fields
1(5)
1.2 Analytic Manifolds
6(3)
1.3 Hensel's Lemma
9(3)
1.4 Differential Forms and Measures
12(4)
1.5 Classification of Compact K-analytic Manifolds
16(2)
1.6 K-analytic Manifolds Associated with Smooth Schemes
18(7)
§ 2 The Theorem of Batyrev--Kontsevich
25(7)
2.1 Calabi--Yau Varieties
25(1)
2.2 Hodge Numbers and Hasse--Weil Zeta Functions
26(4)
2.3 From Complex Numbers to p-adic Numbers
30(2)
§ 3 Igusa's Local Zeta Function
32(23)
3.1 The Local Zeta Function
32(8)
3.2 Denef's Formula
40(3)
3.3 The Topological Zeta Function of Denef--Loeser
43(3)
3.4 The Monodromy Conjecture
46(4)
3.5 Poincare Series
50(5)
2 The Grothendieck Ring of Varieties
55(98)
§ 1 Additive Invariants on Algebraic Varieties
56(11)
1.1 Definition and Examples
56(1)
1.2 The Grothendieck Group of Varieties
57(2)
1.3 Constructible Subsets and Additive Invariants
59(4)
1.4 Piecewise Isomorphisms and Additive Invariants
63(4)
§ 2 Motivic Measures
67(13)
2.1 Definition of Motivic Measures
67(1)
2.2 The Ring Structure on K0(Vars)
68(1)
2.3 Piecewise Trivial Fibrations
69(2)
2.4 Some Classes in K0(Vars)
71(3)
2.5 Spreading-Out and Applications
74(3)
2.6 Variants
77(3)
§ 3 Cohomological Realizations
80(29)
3.1 Grothendieck Rings of Categories
80(5)
3.2 Mixed Hodge Theory and Motivic Measures
85(5)
3.3 Hodge Realization over a Base
90(4)
3.4 Etale Cohomology and Motivic Measures
94(4)
3.5 Etale Realization over a Base
98(5)
3.6 The Crystalline Realization
103(5)
3.7 Motivic Homotopic Realizations
108(1)
§ 4 Localization, Completion, and Modification
109(11)
4.1 Dimensional Filtration
109(1)
4.2 Localization
110(1)
4.3 Completion
111(2)
4.4 A Modified Grothendieck Ring of Varieties
113(7)
§ 5 The Theorem of Bittner
120(13)
5.1 Bittner's Presentation of K0(Vars)
120(7)
5.2 Application to the Construction of Motivic Measures
127(2)
5.3 Motives and Motivic Measures
129(4)
§ 6 The Theorem of Larsen--Lunts and Its Applications
133(20)
6.1 The Theorem of Larsen--Lunts
133(4)
6.2 Other Examples of Motivic Measures
137(2)
6.3 The Cut-and-Paste Property
139(5)
6.4 Zero Divisors in the Grothendieck Ring of Varieties
144(3)
6.5 Algebraically Independent Classes
147(6)
3 Arc Schemes
153(58)
§ 1 Weil Restriction
153(9)
1.1 Reminders on Representability
154(3)
1.2 The Weil Restriction Functor
157(3)
1.3 Representability of a Weil Restriction: The Affine Case
160(1)
1.4 Representability: The General Case
161(1)
§ 2 Jet Schemes
162(6)
2.1 Jet Schemes of a Variety
162(3)
2.2 Truncation Morphisms
165(1)
2.3 Examples
166(2)
§ 3 The Arc Scheme of a Variety
168(20)
3.1 Arcs on a Variety
169(2)
3.2 Relative Representability Properties
171(1)
3.3 Representability of the Functor of Arcs
172(5)
3.4 Base Point and Generic Point of an Arc
177(3)
3.5 Constant Arcs
180(1)
3.6 Renormalization of Arcs
181(2)
3.7 Differential Properties of Jets and Arc Schemes
183(5)
§ 4 Topological Properties of Arc Schemes
188(9)
4.1 Connected Components of Arc Schemes
188(1)
4.2 Irreducible Components of Arc Schemes
189(2)
4.3 Kolchin's Irreducibility Theorem
191(3)
4.4 Application of the Valuative Criterion
194(1)
4.5 Irreducible Components of Constructible Subsets in Arc Spaces
195(2)
§ 5 The Theorem of Grinberg--Kazhdan--Drinfeld
197(14)
5.1 Formal Completion of the Space of Arcs
197(3)
5.2 Weierstrass Theorems for Power Series
200(2)
5.3 Reduction to the Complete Intersection Case
202(2)
5.4 Proof of the Theorem of Grinberg--Kazhdan--Drinfeld
204(3)
5.5 Gabber's Cancellation Theorem and Consequences
207(4)
4 Greenberg Schemes
211(52)
§ 1 Complete Discrete Valuation Rings
212(13)
1.1 Witt Vectors
212(6)
1.2 Complete Discrete Valuation Rings and Their Extensions
218(3)
1.3 The Structure of Complete Discrete Valuation Rings
221(4)
§ 2 The Ring Schemes n
225(15)
2.1 Construction: The Equal Characteristic Case
225(1)
2.2 Construction: The Mixed Characteristic Case
226(8)
2.3 Basic Properties of the Ring Schemes n
234(2)
2.4 The Ideal Schemes mn
236(4)
§ 3 Greenberg Schemes
240(15)
3.1 Greenberg Schemes as Functors
240(6)
3.2 Representability of the Greenberg Schemes
246(2)
3.3 Greenberg Schemes of Formal Schemes
248(3)
3.4 Neron Smoothenings of Formal Schemes
251(2)
3.5 Neron Smoothening and Greenberg Schemes
253(2)
§ 4 Topological Properties of Greenberg Schemes
255(8)
4.1 Irreducible Components of Greenberg Schemes
255(1)
4.2 Constructible Subsets of Greenberg Schemes
256(1)
4.3 Thin Subsets of Greenberg Schemes
257(3)
4.4 Order Functions and Constructible Sets
260(3)
5 Structure Theorems for Greenberg Schemes
263(42)
§ 1 Greenberg Approximation on Formal Schemes
264(13)
1.1 Fitting Ideals
264(1)
1.2 Greenberg Schemes of Smooth Formal Schemes
265(1)
1.3 The Singular Locus of a Formal Scheme
266(4)
1.4 An Application of Hensel's Lemma
270(1)
1.5 Greenberg's Approximation Theorem
271(6)
§ 2 The Structure of the Truncation Morphisms
277(11)
2.1 Principal Homogeneous Spaces and Affine Bundles
277(1)
2.2 Truncation Morphisms and Principal Homogeneous Spaces
278(4)
2.3 The Images of the Truncation Morphisms
282(6)
§ 3 Greenberg Schemes and Morphisms of Formal Schemes
288(17)
3.1 The Jacobian Ideal and the Function ordjacf
288(5)
3.2 Description of the Fibers
293(4)
3.3 Codimension of Constructible Sets in Greenberg Spaces
297(3)
3.4 Example: Contact Loci in Arc Spaces
300(5)
6 Motivic Integration
305(58)
§ 1 Motivic Integration in the Smooth Case
307(4)
1.1 Working with Constructible Sets
307(2)
1.2 The Change of Variables Formula in the Smooth Case
309(2)
§ 2 The Volume of a Constructible Subset of a Greenberg Scheme
311(7)
2.1 What Is a Motivic Volume?
311(1)
2.2 Reduction to the Reduced Flat Case
311(1)
2.3 A Dimensional Lemma
312(2)
2.4 Volume of Thin Constructible Subsets
314(2)
2.5 Existence of the Volume of a Constructible Subset
316(2)
§ 3 Measurable Subsets of Greenberg Schemes
318(15)
3.1 Summable Families in R0
319(1)
3.2 Definition of Measurable Subsets
320(3)
3.3 Existence and Uniqueness of the Volume of Measurable Subsets
323(3)
3.4 Countable Additivity of the Measure μ
326(4)
3.5 Negligible Sets
330(1)
3.6 C-Measurable Subsets of Gr(x)
331(2)
§ 4 Motivic Integrals
333(12)
4.1 Integrable Functions
334(2)
4.2 Direct and Inverse Images of Measurable Subsets
336(4)
4.3 The Change of Variables Formula
340(2)
4.4 An Example: The Blow-Up
342(3)
§ 5 Semi-algebraic Subsets of Greenberg Schemes
345(18)
5.1 Semi-algebraic Subsets
345(2)
5.2 Semi-algebraic Subsets of Greenberg Schemes
347(4)
5.3 Measurability of Semi-algebraic Subsets
351(2)
5.4 Rationality of Motivic Power Series
353(10)
7 Applications
363(1)
§ 1 Kapranov's Motivic Zeta Function
364(1)
1.1 Symmetric Products of Varieties
364(10)
1.2 Definition of Kapranov's Motivic Zeta Function
374(3)
1.3 Motivic Zeta Functions of Curves
377(3)
1.4 Motivic Zeta Functions of Surfaces
380(4)
1.5 Rationality of Kapranov's Zeta Function of Finite Dimensional Motives
384(2)
§ 2 Valuations and the Space of Arcs
386(17)
2.1 Divisorial Valuations and Discrepancies
386(2)
2.2 Valuations Defined by Algebraically Fat Arcs
388(3)
2.3 Minimal Log Discrepancies and the Log Canonical Threshold
391(2)
2.4 Arc Spaces and the Log Canonical Threshold
393(7)
2.5 The Nash Problem
400(3)
§ 3 Motivic Volume and Birational Invariants
403(16)
3.1 Motivic Power Series
403(2)
3.2 The Jacobian Ideal
405(2)
3.3 Motivic Igusa Zeta Functions
407(7)
3.4 Stringy Invariants
414(3)
3.5 The Theorem of Batyrev--Kontsevich
417(2)
§ 4 Denef--Loeser's Zeta Function and the Monodromy Conjecture
419(8)
4.1 Motivic Zeta Functions Associated with Hypersurfaces
419(4)
4.2 The Motivic Nearby Fiber
423(1)
4.3 Lefschetz Numbers of the Monodromy
424(3)
4.4 The Motivic Monodromy Conjecture
427(1)
§ 5 Motivic Invariants of Non-Archimedean Analytic Spaces
427(12)
5.1 Neron Smoothening for Formal R-schemes Formally of Finite Type
428(1)
5.2 Motivic Integration of Volume Forms on Rigid Varieties
429(5)
5.3 The Motivic Serre Invariant
434(1)
5.4 Comparison with p-adic Integration
435(2)
5.5 The Trace Formula
437(2)
§ 6 Motivic Zeta Functions of Formal Schemes and Analytic Spaces
439(12)
6.1 Definition of the Motivic Zeta Function
439(1)
6.2 Bounded Differential Forms
440(1)
6.3 Resolution of Singularities for Formal Schemes
441(3)
6.4 Neron Smoothening After Ramification
444(2)
6.5 A Formula for the Motivic Zeta Function
446(3)
6.6 Comparison with Denef and Loeser's Motivic Zeta Function
449(2)
6.7 Motivic Zeta Functions of Calabi--Yau Varieties
451(1)
§ 7 Motivic Serre Invariants of Algebraic Varieties
451(1)
7.1 Weak Neron Models of Algebraic Varieties
452(3)
7.2 Motivic Integrals and Motivic Serre Invariants for Smooth Algebraic Varieties
455(3)
7.3 Motivic Serre Invariants of Open and Singular Varieties
458(2)
7.4 The Trace Formula
460
Correction to: Motivic Integration
1(464)
Appendix
465(34)
§ 1 Constructibility in Algebraic Geometry
465(4)
1.1 Constructible Subsets of a Scheme
465(2)
1.2 The Constructible Topology
467(1)
1.3 Constructible Subsets of Projective Limits
468(1)
§ 2 Birational Geometry
469(9)
2.1 Blow-Ups
469(1)
2.2 Resolution of Singularities
470(1)
2.3 Weak Factorization Theorem
471(1)
2.4 Canonical Divisors and Resolutions
472(2)
2.5 K-equivalence
474(2)
2.6 A Birational Cancellation Lemma
476(2)
§ 3 Formal and Non-Archimedean Geometry
478(21)
3.1 Formal Schemes
478(5)
3.2 Morphisms of Finite Type and Morphisms Formally of Finite Type
483(3)
3.3 Smoothness and Differentials
486(2)
3.4 Formal Schemes over a Complete Discrete Valuation Ring
488(3)
3.5 Non-Archimedean Analytic Spaces
491(8)
Bibliography 499(20)
Index 519