Motivic integration. ([2018])

• Record Type: Book
• Further Information: Note: Antoine Chambert-Loir, Johannes Nicaise, Julien Sebag.
• Authors: Chambert-Loir, Antoine; Nicaise, Johannes; Sebag, Julien
• Contents: Intro; Introduction; Contents; Chapter 1. Prologue: p-Adic Integration; 1. Analytic Manifolds; 1.1. Local Fields; 1.2. Analytic Manifolds; 1.3. Hensel's Lemma; 1.4. Differential Forms and Measures; 1.5. Classification of Compact K-analytic Manifolds; 1.6. K-analytic Manifolds Associated with Smooth Schemes; 2. The Theorem of Batyrev-Kontsevich; 2.1. Calabi-Yau Varieties; 2.2. Hodge Numbers and Hasse-Weil Zeta Functions; 2.3. From Complex Numbers to p-adic Numbers; 3. Igusa's Local Zeta Function; 3.1. The Local Zeta Function; 3.2. Denef's Formula 3.3. The Topological Zeta Function of Denef-Loeser3.4. The Monodromy Conjecture; 3.5. Poincaré Series; Chapter 2. The Grothendieck Ring of Varieties; 1. Additive Invariants on Algebraic Varieties; 1.1. Definition and Examples; 1.2. The Grothendieck Group of Varieties; 1.3. Constructible Subsets and Additive Invariants; 1.4. Piecewise Isomorphisms and Additive Invariants; 2. Motivic Measures; 2.1. Definition of Motivic Measures; 2.2. The Ring Structure on K0(VarS); 2.3. Piecewise Trivial Fibrations; 2.4. Some Classes in K0(VarS); 2.5. Spreading-Out and Applications; 2.6. Variants 3. Cohomological Realizations3.1. Grothendieck Rings of Categories; 3.2. Mixed Hodge Theory and Motivic Measures; 3.3. Hodge Realization over a Base; 3.4. Étale Cohomology and Motivic Measures; 3.5. Étale Realization over a Base; 3.6. The Crystalline Realization; 3.7. Motivic Homotopic Realizations; 4. Localization, Completion, and Modification; 4.1. Dimensional Filtration; 4.2. Localization; 4.3. Completion; 4.4. A Modified Grothendieck Ring of Varieties; 5. The Theorem of Bittner; 5.1. Bittner's Presentation of K0(VarS); 5.2. Application to the Construction of Motivic Measures 5.3. Motives and Motivic Measures 6. The Theorem of Larsen-Lunts and Its Applications; 6.1. The Theorem of Larsen-Lunts; 6.2. Other Examples of Motivic Measures; 6.3. The Cut-and-Paste Property; 6.4. Zero Divisors in the Grothendieck Ring of Varieties; 6.5. Algebraically Independent Classes; Chapter 3. Arc Schemes; 1. Weil Restriction; 1.1. Reminders on Representability; 1.2. The Weil Restriction Functor; 1.3. Representability of a Weil Restriction: The Affine Case; 1.4. Representability: The General Case; 2. Jet Schemes; 2.1. Jet Schemes of a Variety; 2.2. Truncation Morphisms 2.3. Examples 3. The Arc Scheme of a Variety; 3.1. Arcs on a Variety; 3.2. Relative Representability Properties; 3.3. Representability of the Functor of Arcs; 3.4. Base Point and Generic Point of an Arc; 3.5. Constant Arcs; 3.6. Renormalization of Arcs; 3.7. Differential Properties of Jets and Arc Schemes; 4. Topological Properties of Arc Schemes; 4.1. Connected Components of Arc Schemes; 4.2. Irreducible Components of Arc Schemes; 4.3. Kolchin's Irreducibility Theorem; 4.4. Application of the Valuative Criterion; 4.5. Irreducible Components of Constructible Subsets in Arc Spaces
• Publisher Details: New York, NY : Birkhäuser
• Publication Date: 2018
• Extent: 1 online resource
• Subjects: 516.3/5; Mathematics; Motives (Mathematics); Geometry, algebraic; MATHEMATICS / Geometry / General; Mathematics -- Algebra -- Abstract; Algebraic topology; Mathematics -- Geometry -- Algebraic; Algebraic geometry; Electronic books
• Languages: English
• ISBNs: 9781493978878
• Related ISBNs: 149397887X; 9781493978854; 1493978853
• Notes: Note: Includes bibliographical references and index.; Note: Online resource; title from PDF title page (EBSCO, viewed September 19, 2018).
• Access Rights: Legal Deposit; Only available on premises controlled by the deposit library and to one user at any one time; The Legal Deposit Libraries (Non-Print Works) Regulations (UK).
• Access Usage: Restricted: Printing from this resource is governed by The Legal Deposit Libraries (Non-Print Works) Regulations (UK) and UK copyright law currently in force.