Zariski Geometries: Geometry from the Logician's Point of View (London Mathematical Society Lecture Note Series, Series Number 360)

This Book Presents Methods And Results From The Theory Of Zariski Structures And Discusses Their Applications In Geometry As Well As Various Other Mathematical Fields. Its Logical Approach Helps Us Understand Why Algebraic Geometry Is So Fundamental Throughout Mathematics And Why The Extension To Noncommutative Geometry, Which Has Been Forced By Recent Developments In Quantum Physics, Is Both Natural And Necessary. Beginning With A Crash Course In Model Theory, This Book Will Suit Not Only Model Theorists But Also Readers With A More Classical Geometric Background--provided By Publisher. Machine Generated Contents Note: 1. Introduction; 2. Topological Structures; 3. Noetherian Zariski Structures; 4. Classification Results; 5. Non-classical Zariski Geometries; 6. Analytic Zariski Geometries; A. Basic Model Theory; B. Geometric Stability Theory; References; Index. Boris Zilber. Includes Bibliographical References (p. 207-209) And Index. Title......Page 4 Copyright......Page 5 Dedication......Page 6 Contents......Page 8 Acknowledgments......Page 12 1.1 Introduction......Page 14 1.2 About model theory......Page 20 2.1 Basic notions......Page 25 2.2 Specialisations......Page 27 2.2.1 Universal specialisations......Page 30 2.2.2 Infinitesimal neighbourhoods......Page 32 2.2.3 Continuous and differentiable function......Page 35 3.1.1 Good dimension......Page 38 3.1.2 Zariski structures......Page 39 3.2.1 Elimination of quantifiers......Page 40 3.3 One-dimensional case......Page 43 3.4.1 Algebraic varieties and orbifolds over algebraically closed fields......Page 48 3.4.2 Compact complex manifolds......Page 49 3.4.3 Proper varieties of rigid analytic geometry......Page 51 3.4.4 Zariski structures living in differentially closed fields......Page 52 3.5.1 Pre-smoothness......Page 53 3.5.2 Coverings in structures with dimension......Page 56 3.5.3 Elementary extensions of Zariski structures......Page 57 3.6.1 Coverings in pre-smooth structures......Page 63 3.6.2 Multiplicities......Page 66 3.6.3 Elements of intersection theory......Page 70 3.6.4 Local isomorphisms......Page 72 3.7 Getting new Zariski sets......Page 75 3.8 Curves and their branches......Page 82 4.1 Getting a group......Page 91 4.1.1 Composing branches of curves......Page 92 4.1.2 Pre-group of jets......Page 95 4.2 Getting a field......Page 101 4.3.1 Projective spaces as Zariski structures......Page 106 4.3.2 Completeness......Page 107 4.3.3 Intersection theory in projective spaces......Page 108 4.3.4 Generalised Bezout and Chow theorems......Page 110 4.4.1 Main theorem......Page 113 4.4.2 Meromorphic functions on a Zariski set......Page 114 4.4.3 Simple Zariski groups are algebraic......Page 116 5.1 Non-algebraic Zariski geometries......Page 118 5.2.2 Semi-definable functions on PN......Page 122 5.2.4 Representation of A......Page 124 5.2.5 Metric limit......Page 128 5.3 From quantum algebras to Zariski structures......Page 133 5.3.1 Algebras at roots of unity......Page 135 5.3.2 Examples......Page 138 5.3.3 Definable sets and Zariski properties......Page 147 6.1 Definition and basic properties......Page 150 6.1.1 Closed and projective sets......Page 151 6.1.2 Analytic subsets......Page 152 6.2 Compact analytic Zariski structures......Page 153 6.3 Model theory of analytic Zariski structures......Page 157 6.4 Specialisations in analytic Zariski structures......Page 166 6.5.1 Covers of algebraic varieties......Page 168 6.5.2 Hard examples......Page 172 A.1 Languages and structures......Page 176 A.2 Compactness theorem......Page 179 A.3 Existentially closed structures......Page 183 A.4 Complete and categorical theories......Page 185 A.4.1 Types in complete theories......Page 188 A.4.2 Spaces of types and saturated models......Page 190 A.4.3 Categoricity in uncountable powers......Page 195 B.1 Algebraic closure in abstract structures......Page 198 B.1.1 Pre-geometry and geometry of a minimal structure......Page 199 B.1.2 Dimension notion in strongly minimal structures......Page 202 B.1.3 Macro- and micro-geometries on a strongly minimal structure......Page 207 B.2.1 Trichotomy conjecture......Page 213 B.2.2 Hrushovski’s construction of new stable structures......Page 215 B.2.3 Pseudo-exponentiation......Page 218 References......Page 220 Index......Page 223 Title 4 Copyright 5 Dedication 6 Contents 8 Acknowledgments 12 1 Introduction 14 1.1 Introduction 14 1.2 About model theory 20 2 Topological structures 25 2.1 Basic notions 25 2.2 Specialisations 27 2.2.1 Universal specialisations 30 2.2.2 Infinitesimal neighbourhoods 32 2.2.3 Continuous and differentiable function 35 3 Noetherian Zariski structures 38 3.1 Topological structures with good dimension notion 38 3.1.1 Good dimension 38 3.1.2 Zariski structures 39 3.2 Model theory of Zariski structures 40 3.2.1 Elimination of quantifiers 40 3.2.2 Morley rank 43 3.3 One-dimensional case 43 3.4 Basic examples 48 3.4.1 Algebraic varieties and orbifolds over algebraically closed fields 48 3.4.2 Compact complex manifolds 49 3.4.3 Proper varieties of rigid analytic geometry 51 3.4.4 Zariski structures living in differentially closed fields 52 3.5 Further geometric notions 53 3.5.1 Pre-smoothness 53 3.5.2 Coverings in structures with dimension 56 3.5.3 Elementary extensions of Zariski structures 57 3.6 Non-standard analysis 63 3.6.1 Coverings in pre-smooth structures 63 3.6.2 Multiplicities 66 3.6.3 Elements of intersection theory 70 3.6.4 Local isomorphisms 72 3.7 Getting new Zariski sets 75 3.8 Curves and their branches 82 4 Classification results 91 4.1 Getting a group 91 4.1.1 Composing branches of curves 92 4.1.2 Pre-group of jets 95 4.2 Getting a field 101 4.3 Projective spaces over a Z-field 106 4.3.1 Projective spaces as Zariski structures 106 4.3.2 Completeness 107 4.3.3 Intersection theory in projective spaces 108 4.3.4 Generalised Bezout and Chow theorems 110 4.4 The classification theorem 113 4.4.1 Main theorem 113 4.4.2 Meromorphic functions on a Zariski set 114 4.4.3 Simple Zariski groups are algebraic 116 5 Non-classical Zariski geometries 118 5.1 Non-algebraic Zariski geometries 118 5.2 Case study 122 5.2.1 The N-cover of the affine line 122 5.2.2 Semi-definable functions on PN 122 5.2.3 Space of semi-definable functions 124 5.2.4 Representation of A 124 5.2.5 Metric limit 128 5.3 From quantum algebras to Zariski structures 133 5.3.1 Algebras at roots of unity 135 5.3.2 Examples 138 5.3.3 Definable sets and Zariski properties 147 6 Analytic Zariski geometries 150 6.1 Definition and basic properties 150 6.1.1 Closed and projective sets 151 6.1.2 Analytic subsets 152 6.2 Compact analytic Zariski structures 153 6.3 Model theory of analytic Zariski structures 157 6.4 Specialisations in analytic Zariski structures 166 6.5 Examples 168 6.5.1 Covers of algebraic varieties 168 6.5.2 Hard examples 172 A Basic model theory 176 A.1 Languages and structures 176 A.2 Compactness theorem 179 A.3 Existentially closed structures 183 A.4 Complete and categorical theories 185 A.4.1 Types in complete theories 188 A.4.2 Spaces of types and saturated models 190 A.4.3 Categoricity in uncountable powers 195 B Elements of geometric stability theory 198 B.1 Algebraic closure in abstract structures 198 B.1.1 Pre-geometry and geometry of a minimal structure 199 B.1.2 Dimension notion in strongly minimal structures 202 B.1.3 Macro- and micro-geometries on a strongly minimal structure 207 B.2 Trichotomy conjecture 213 B.2.1 Trichotomy conjecture 213 B.2.2 Hrushovski’s construction of new stable structures 215 B.2.3 Pseudo-exponentiation 218 References 220 Index 223