Verfügbarkeit: Lectures on algebraic geometry

Lectures on algebraic geometry: 1 Sheaves, cohomology of sheaves, and applications to Riemann surfaces

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1. Verfasser:	Harder, Günter (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	Wiesbaden Vieweg 2008
Ausgabe:	1. ed.
Schriftenreihe:	Aspects of mathematics E ; 35
Online-Zugang:	Inhaltsverzeichnis Inhaltsverzeichnis
Beschreibung:	XIV, 290 S. graph. Darst.
ISBN:	9783528031367

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Datensatz im Suchindex

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adam_text	Contents Preface v Introduction xiii 1 Categories, Products, Protective and Inductive Limits 1 1.1 The Notion of a Category and Examples ................. 1 1.2 Functors ................................... 3 1.3 Products, Projective Limits and Direct Limits in a Category ..... 4 1.3.1 The Projective Limit .......................... 4 1.3.2 The Yoneda Lemma .......................... 6 1.3.3 Examples ................................ 6 1.3.4 Representable Functors ........................ 8 1.3.5 Direct Limits .............................. 9 1.4 Exercises .................................. 10 2 Basic Concepts of Homological Algebra 11 2.1 The Category Modr of Г -modules .................... 11 2.2 More Functors ............................... 13 2.2.1 Invariants, Coinvariants and Exactness ................ 13 2.2.2 The First Cohomology Group ..................... 15 2.2.3 Some Notation ............................. 16 2.2.4 Exercises ................................ 17 2.3 The Derived Functors ........................... 19 2.3.1 The Simple Principle .......................... 20 2.3.2 Functoriality .............................. 22 2.3.3 Other Resolutions ........................... 24 2.3.4 Injective Resolutions of Short Exact Sequences ........... 24 A Fundamental Remark ........................ 26 The Cohomology and the Long Exact Sequence ........... 27 The Homology of Groups ....................... 27 2.4 The Functors Ext and Tor ......................... 28 2.4.1 The Functor Ext ............................ 28 2.4.2 The Derived Functor for the Tensor Product ............ 30 2.4.3 Exercise ................................. 32 3 Sheaves 35 3.1 Presheaves and Sheaves .......................... 35 3.1.1 What is a Presheaf? .......................... 35 3.1.2 A Remark about Products and Presheaf ............... 36 3.1.3 What is a Sheaf? ........................... 36 3.1.4 Examples ................................ 38 3.2 Manifolds as Locally Ringed Spaces ................... 39 3.2.1 What Are Manifolds? ......................... 39 3.2.2 Examples and Exercise ......................... 41 3.3 Stalks and Sheafification .......................... 45 3.3.1 Stalks .................................. 45 3.3.2 The Process of Sheafification of a Presheaf .............. 46 3.4 The Functors ƒ* and ƒ.......................... 47 3.4.1 The Adjunction Formula ........................ 48 3.4.2 Extensions and Restrictions ...................... 49 3.5 Constructions of Sheaves ......................... 49 Cohomology of Sheaves 51 4.1 Examples .................................. 51 4.1.1 Sheaves on Riemann surfaces ..................... 51 4.1.2 Cohomology of the Circle ....................... 54 4.2 The Derived Functor ............................ 55 4.2.1 Injective Sheaves and Derived Functors ............... 55 4.2.2 A Direct Definition of Я1 ....................... 56 4.3 Fiber Bundles and Non Abelian H1 ................... 59 4.3.1 Fibrations ................................ 59 Fibre Bundle .............................. 59 Vector Bundles ............................. 60 4.3.2 Non- Abelian Я1 ............................ 61 4.3.3 The Reduction of the Structure Group ................ 62 Orientation ............................... 62 Local Systems ............................. 63 Isomorphism Classes of Local Systems ................ 64 Principal G-bundels .......................... 64 4.4 Fundamental Properties of the Cohomology of Sheaves ........ 65 4.4.1 Introduction .............................. 65 4.4.2 The Derived Functor to ƒ ...................... 66 4.4.3 Functorial Properties of the Cohomology .............. 67 4.4.4 Paracompact Spaces .......................... 69 4.4.5 Applications .............................. 75 Cohomology of Spheres ........................ 75 Orientations ............................... 76 Compact Oriented Surfaces ...................... 77 4.5 Cech Cohomology of Sheaves ....................... 77 4.5.1 The Čech-Complex ........................... 77 4.5.2 The Čech Resolution of a Sheaf .................... 80 4.6 Spectral Sequences ............................. 82 4.6.1 Introduction .............................. 82 4.6.2 The Vertical Filtration ......................... 87 4.6.3 The Horizontal Filtration ....................... 93 Two Special Cases ........................... 94 Applications of Spectral Sequences .................. 95 4.6.4 The Derived Category ......................... 97 The Composition Rule ......................... 100 Exact Triangles ............................. 101 4.6.5 The Spectral Sequence of a Fibration ................ 102 Sphere Bundles an Euler Characteristic ............... 103 4.6.6 Čech Complexes and the Spectral Sequence ............. 104 A Criterion for Degeneration ..................... 106 An Application to Product Spaces .................. 108 4.6.7 The Cup Product ............................ 110 4.6.8 Example: Cup Product for the Comology of Tori .......... 114 A Connection to the Cohomology of Groups ............ 115 4.6.9 An Excursion into Homotopy Theory ................ 116 4.7 Cohomology with Compact Supports .................-. 119 4.7.1 The Definition ............................. 119 4.7.2 An Example for Cohomology with Compact Supports ....... 120 The Cohomology with Compact Supports for Open Balls ..... 120 Formulae for Cup Products ...................... 122 4.7.3 The Fundamental Class ........................ 124 4.8 Cohomology of Manifolds ......................... 125 4.8.1 Local Systems ............................. 125 4.8.2 Čech Resolutions of Local Systems .................. 126 4.8.3 Čech Coresolution of Local Systems ................. 128 4.8.4 Poincaré Duality ............................ 131 4.8.5 The Cohomology in Top Degree and the Homology ......... 137 4.8.6 Some Remarks on Singular Homology ................ 139 4.8.7 Cohomology with Compact Support and Embeddings ....... 140 4.8.8 The Fundamental Class of a Submanifold .............. 142 4.8.9 Cup Product and Intersections .................... 143 4.8.10 Compact oriented Surfaces ...................... 145 4.8.11 The Cohomology Ring of Pn(C) ................... 146 4.9 The Lefschetz Fixed Point Formula .................... 146 4.9.1 The Euler Characteristic of Manifolds ................ 148 4.10 The de Rham and the Dolbeault Isomorphism ............. 149 4.10.1 The Cohomology of Flat Bundles on Real Manifolds ........ 149 The Product Structure on the de Rham Cohomology ........ 152 The de Rham Isomorphism and the fundamental class ....... 153 4.10.2 Cohomology of Holomorphic Bundles on Complex Manifolds . . . 155 The Tangent Bundle ......................... 155 The Bundle tl™ ............................ 157 4.10.3 Chern Classes ............................. 159 The Line Bundles O¥n{c)(k) ..................... 162 4.11 Hodge Theory ............................... 163 4.11.1 Hodge Theory on Real Manifolds ................... 163 4.11.2 Hodge Theory on Complex Manifolds ................ 168 Some Linear Algebra ......................... 168 Kahler Manifolds and their Cohomology ............... 171 The Cohomology of Holomorphic Vector Bundles ......... 174 Serre Duality .............................. 175 4.11.3 Hodge Theory on Tori ........................ 176 5 Compact Riemann surfaces and Abelian Varieties 179 5.1 Compact Riemann Surfaces ........................ 179 5.1.1 Introduction .............................. 179 5.1.2 The Hodge Structure on ff1 (S1,C) .................. 180 5.1.3 Cohomology of Holomorphic Bundles ................. 185 5.1.4 The Theorem of Riemann- Roch.................... 191 On the Picard Group ......................... 191 Exercises ................................ 192 The Theorem of Riemann-Roch .................... 193 5.1.5 The Algebraic Duality Pairing .................... 194 5.1.6 Riemann Surfaces of Low Genus ................... 196 5.1.7 The Algebraicity of Riemann Surfaces ................ 197 From a Riemann Surface to Function Fields ............. 197 The reconstruction of S from К ................... 202 Connection to Algebraic Geometry .................. 209 Elliptic Curves ............................. 211 5.1.8 Géométrie Analytique et Géométrie Algébrique - GAGA...... 212 5.1.9 Comparison of Two Pairings ..................... 215 5.1.10 The Jacobian of a Compact Riemann Surface ............ 217 5.1.11 The Classical Version of Abel s Theorem .............. 218 5.1.12 Riemann Period Relations ...................... 222 5.2 Line Bundles on Complex Tori ...................... 223 5.2.1 Construction of Line Bundles ..................... 223 The Poincaré Bundle .......................... 229 Universality of AT ............................ 230 5.2.2 Homomorphisms Between Complex Tori ............... 232 The Neron Severi group and Нот(Л,Лу) ............... 234 The construction of Φ starting from a line bundle ......... 235 5.2.3 The Self Duality of the Jacobian ................... 236 5.2.4 Ample Line Bundles and the Algebraicity of the Jacobian ..... 237 The Kodaira Embedding Theorem .................. 237 The Spaces of Sections ......................... 239 5.2.5 The Siegel Upper Half Space ..................... 240 5.2.6 Riemann-Theta Functions ....................... 243 5.2.7 Projective embeddings of abelian varieties .............. 248 5.2.8 Degeneration of Abelian Varieties ................... 250 The Case of Genus 1.......................... 251 The Algebraic Approach ........................ 261 5.3 Towards the Algebraic Theory ...................... 263 5.3.1 Introduction .............................. 263 The Algebraic Definition of the Neron-Severi Group ........ 264 The Algebraic Definition of the Intersection Numbers ....... 264 The Study of some Special Neron-Severi groups .......... 266 Xl 5.3.2 The Structure of End( J) ........................270 The Rosati Involution .........................270 A Trace Formula ............................272 The Fundamental Class [S] of S under the Abel Map .......275 5.3.3 The Ring of Correspondences ..................... 276 5.3.4 An Algebraic Substitute for the Cohomology ............277 Bibliography 281 Index 284 GIINTER HARDER LECTURES ON ALGEBRAIC GEOMETRY I SHEAVES, COHOMOLOGY OF SHEAVES, AND APPLICATIONS TO RIEMANN SURFACES VIEWEG CONTENTS PREFACE V INTRODUCTION XIII 1 CATEGORIES, PRODUCTS, PROJECTIVE AND INDUCTIVE LIMITS 1 1.1 THE NOTION OF A CATEGORY AND EXAMPLES 1 1.2 FUNCTORS 3 1.3 PRODUCTS, PROJECTIVE LIMITS AND DIRECT LIMITS IN A CATEGORY 4 1.3.1 THE PROJECTIVE LIMIT 4 1.3.2 THE YONEDA LEMMA 6 1.3.3 EXAMPLES 6 1.3.4 REPRESENTABLE FUNCTORS 8 1.3.5 DIRECT LIMITS 9 1.4 EXERCISES 10 2 BASIC CONCEPTS OF HOMOLOGICAL ALGEBRA 11 2.1 THE CATEGORY MODR OF F-MODULES 11 2.2 MORE FUNCTORS 13 2.2.1 INVARIANTS, COINVARIANTS AND EXACTNESS 13 2.2.2 THE FIRST COHOMOLOGY GROUP 15 2.2.3 SOME NOTATION 16 2.2.4 EXERCISES 17 2.3 THE DERIVED FUNCTORS 19 2.3.1 THE SIMPLE PRINCIPLE 20 2.3.2 FUNCTORIALITY 22 2.3.3 OTHER RESOLUTIONS 24 2.3.4 INJECTIVE RESOLUTIONS OF SHORT EXACT SEQUENCES 24 A FUNDAMENTAL REMARK 26 THE COHOMOLOGY AND THE LONG EXACT SEQUENCE 27 THE HOMOLOGY OF GROUPS 27 2.4 THE FUNCTORS EXT AND TOR 28 2.4.1 THE FUNCTOR EXT 28 2.4.2 THE DERIVED FUNCTOR FOR THE TENSOR PRODUCT 30 2.4.3 EXERCISE 32 3 SHEAVES 35 3.1 PRESHEAVES AND SHEAVES 35 3.1.1 WHAT IS A PRESHEAF? 35 3.1.2 A REMARK ABOUT PRODUCTS AND PRESHEAF 36 3.1.3 WHAT IS A SHEAF? 36 3.1.4 EXAMPLES 38 3.2 MANIFOLDS AS LOCALLY RINGED SPACES 39 I CONTENTS 3.2.1 WHAT ARE MANIFOLDS? 39 3.2.2 EXAMPLES AND EXERCISE 41 3.3 STALKS AND SHEAFIFICATION 45 3.3.1 STALKS 45 3.3.2 THE PROCESS OF SHEAFIFICATION OF A PRESHEAF 46 3.4 THE FUNCTORS /, AND /* 47 3.4.1 THE ADJUNCTION FORMULA 48 3.4.2 EXTENSIONS AND RESTRICTIONS 49 3.5 CONSTRUCTIONS OF SHEAVES 49 COHOMOLOGY OF SHEAVES 51 4.1 EXAMPLES 51 4.1.1 SHEAVES ON RIEMANN SURFACES 51 4.1.2 COHOMOLOGY OF THE CIRCLE 54 4.2 THE DERIVED FUNCTOR 55 4.2.1 INJECTIVE SHEAVES AND DERIVED FUNCTORS 55 4.2.2 A DIRECT DEFINITION OF H 1 56 4.3 FIBER BUNDLES AND NON ABELIAN H 1 59 4.3.1 FIBRATIONS 59 FIBRE BUNDLE 59 VECTOR BUNDLES 60 4.3.2 NON-ABELIAN H 1 61 4.3.3 THE REDUCTION OF THE STRUCTURE GROUP 62 ORIENTATION 62 LOCAL SYSTEMS 63 ISOMORPHISM CLASSES OF LOCAL SYSTEMS 64 PRINCIPAL G-BUNDELS 64 4.4 FUNDAMENTAL PROPERTIES OF THE COHOMOLOGY OF SHEAVES 65 4.4.1 INTRODUCTION 65 4.4.2 THE DERIVED FUNCTOR TO /, 66 4.4.3 FUNCTORIAL PROPERTIES OF THE COHOMOLOGY 67 4.4.4 PARACOMPACT SPACES 69 4.4.5 APPLICATIONS 75 COHOMOLOGY OF SPHERES 75 ORIENTATIONS 76 COMPACT ORIENTED SURFACES 77 4.5 CECH COHOMOLOGY OF SHEAVES 77 4.5.1 THE CECH-COMPLEX 77 4.5.2 THE CECH RESOLUTION OF A SHEAF 80 4.6 SPECTRAL SEQUENCES 82 4.6.1 INTRODUCTION 82 4.6.2 THE VERTICAL FILTRATION 87 4.6.3 THE HORIZONTAL FILTRATION 93 TWO SPECIAL CASES 94 APPLICATIONS OF SPECTRAL SEQUENCES 95 4.6.4 THE DERIVED CATEGORY 97 THE COMPOSITION RULE 100 EXACT TRIANGLES 101 4.6.5 THE SPECTRAL SEQUENCE OF A FIBRATION 102 SPHERE BUNDLES AN EULER CHARACTERISTIC 103 4.6.6 CECH COMPLEXES AND THE SPECTRAL SEQUENCE 104 A CRITERION FOR DEGENERATION 106 AN APPLICATION TO PRODUCT SPACES 108 4.6.7 THE CUP PRODUCT 110 4.6.8 EXAMPLE: CUP PRODUCT FOR THE COMOLOGY OF TORI 114 A CONNECTION TO THE COHOMOLOGY OF GROUPS 115 4.6.9 AN EXCURSION INTO HOMOTOPY THEORY 116 4.7 COHOMOLOGY WITH COMPACT SUPPORTS 119 4.7.1 THE DEFINITION 119 4.7.2 AN EXAMPLE FOR COHOMOLOGY WITH COMPACT SUPPORTS 120 THE COHOMOLOGY WITH COMPACT SUPPORTS FOR OPEN BALLS 120 FORMULAE FOR CUP PRODUCTS 122 4.7.3 THE FUNDAMENTAL CLASS 124 4.8 COHOMOLOGY OF MANIFOLDS 125 4.8.1 LOCAL SYSTEMS 125 4.8.2 CECH RESOLUTIONS OF LOCAL SYSTEMS 126 4.8.3 CECH CORESOLUTION OF LOCAL SYSTEMS 128 4.8.4 POINCARE DUALITY 131 4.8.5 THE COHOMOLOGY IN TOP DEGREE AND THE HOMOLOGY 137 4.8.6 SOME REMARKS ON SINGULAR HOMOLOGY 139 4.8.7 COHOMOLOGY WITH COMPACT SUPPORT AND EMBEDDINGS 140 4.8.8 THE FUNDAMENTAL CLASS OF A SUBMANIFOLD 142 4.8.9 CUP PRODUCT AND INTERSECTIONS 143 4.8.10 COMPACT ORIENTED SURFACES 145 4.8.11 THE COHOMOLOGY RING OF P N (C) 146 4.9 THE LEFSCHETZ FIXED POINT FORMULA 146 4.9.1 THE EULER CHARACTERISTIC OF MANIFOLDS 148 4.10 THE DE RHAM AND THE DOLBEAULT ISOMORPHISM 149 4.10.1 THE COHOMOLOGY OF FLAT BUNDLES ON REAL MANIFOLDS 149 THE PRODUCT STRUCTURE ON THE DE RHAM COHOMOLOGY 152 THE DE RHAM ISOMORPHISM AND THE FUNDAMENTAL CLASS 153 4.10.2 COHOMOLOGY OF HOLOMORPHIC BUNDLES ON COMPLEX MANIFOLDS . . . 155 THE TANGENT BUNDLE 155 THE BUNDLE &$ 157 4.10.3 CHERN CLASSES 159 THE LINE BUNDLES O PN(C ){K) 162 4.11 HODGE THEORY 163 4.11.1 HODGE THEORY ON REAL MANIFOLDS 163 4.11.2 HODGE THEORY ON COMPLEX MANIFOLDS 168 SOME LINEAR ALGEBRA 168 KAHLER MANIFOLDS AND THEIR COHOMOLOGY 171 THE COHOMOLOGY OF HOLOMORPHIC VECTOR BUNDLES 174 SERRE DUALITY 175 4.11.3 HODGE THEORY ON TORI 176 CONTENTS COMPACT RIEMANN SURFACES AND ABELIAN VARIETIES 179 5.1 COMPACT RIEMANN SURFACES 179 5.1.1 INTRODUCTION 179 5.1.2 THE HODGE STRUCTURE ON IF 1 (5,0) 180 5.1.3 COHOMOLOGY OF HOLOMORPHIC BUNDLES 185 5.1.4 THE THEOREM OF RIEMANN-ROCH 191 ON THE PICARD GROUP 191 EXERCISES 192 THE THEOREM OF RIEMANN-ROCH 193 5.1.5 THE ALGEBRAIC DUALITY PAIRING 194 5.1.6 RIEMANN SURFACES OF LOW GENUS 196 5.1.7 THE ALGEBRAICITY OF RIEMANN SURFACES 197 FROM A RIEMANN SURFACE TO FUNCTION FIELDS 197 THE RECONSTRUCTION OF S FROM K 202 CONNECTION TO ALGEBRAIC GEOMETRY 209 ELLIPTIC CURVES 211 5.1.8 GEOMETRIE ANALYTIQUE ET GEOMETRIE ALGEBRIQUE - GAGA 212 5.1.9 COMPARISON OF TWO PAIRINGS 215 5.1.10 THE JACOBIAN OF A COMPACT RIEMANN SURFACE 217 5.1.11 THE CLASSICAL VERSION OF ABEL S THEOREM 218 5.1.12 RIEMANN PERIOD RELATIONS 222 5.2 LINE BUNDLES ON COMPLEX TORI 223 5.2.1 CONSTRUCTION OF LINE BUNDLES 223 THE POINCARE BUNDLE 229 UNIVERSALITY OF M 230 5.2.2 HOMOMORPHISMS BETWEEN COMPLEX TORI 232 THE NERON SEVERI GROUP AND HOM(,4,.4 V ) 234 THE CONSTRUCTION OF F STARTING FROM A LINE BUNDLE 235 5.2.3 THE SELF DUALITY OF THE JACOBIAN 236 5.2.4 AMPLE LINE BUNDLES AND THE ALGEBRAICITY OF THE JACOBIAN 237 THE KODAIRA EMBEDDING THEOREM 237 THE SPACES OF SECTIONS 239 5.2.5 THE SIEGEL UPPER HALF SPACE 240 5.2.6 RIEMANN-THETA FUNCTIONS 243 5.2.7 PROJECTIVE EMBEDDINGS OF ABELIAN VARIETIES 248 5.2.8 DEGENERATION OF ABELIAN VARIETIES 250 THE CASE OF GENUS 1 251 THE ALGEBRAIC APPROACH 261 5.3 TOWARDS THE ALGEBRAIC THEORY 263 5.3.1 INTRODUCTION 263 THE ALGEBRAIC DEFINITION OF THE NERON-SEVERI GROUP 264 THE ALGEBRAIC DEFINITION OF THE INTERSECTION NUMBERS 264 THE STUDY OF SOME SPECIAL NERON-SEVERI GROUPS 266 5.3.2 THE STRUCTURE OF END( J) 270 THE ROSATI INVOLUTION 270 A TRACE FORMULA 272 THE FUNDAMENTAL CLASS [S] OF S UNDER THE ABEL MAP 275 5.3.3 THE RING OF CORRESPONDENCES 276 5.3.4 AN ALGEBRAIC SUBSTITUTE FOR THE COHOMOLOGY 277 BIBLIOGRAPHY 281 INDEX 284
adam_txt	Contents Preface v Introduction xiii 1 Categories, Products, Protective and Inductive Limits 1 1.1 The Notion of a Category and Examples . 1 1.2 Functors . 3 1.3 Products, Projective Limits and Direct Limits in a Category . 4 1.3.1 The Projective Limit . 4 1.3.2 The Yoneda Lemma . 6 1.3.3 Examples . 6 1.3.4 Representable Functors . 8 1.3.5 Direct Limits . 9 1.4 Exercises . 10 2 Basic Concepts of Homological Algebra 11 2.1 The Category Modr of Г -modules . 11 2.2 More Functors . 13 2.2.1 Invariants, Coinvariants and Exactness . 13 2.2.2 The First Cohomology Group . 15 2.2.3 Some Notation . 16 2.2.4 Exercises . 17 2.3 The Derived Functors . 19 2.3.1 The Simple Principle . 20 2.3.2 Functoriality . 22 2.3.3 Other Resolutions . 24 2.3.4 Injective Resolutions of Short Exact Sequences . 24 A Fundamental Remark . 26 The Cohomology and the Long Exact Sequence . 27 The Homology of Groups . 27 2.4 The Functors Ext and Tor . 28 2.4.1 The Functor Ext . 28 2.4.2 The Derived Functor for the Tensor Product . 30 2.4.3 Exercise . 32 3 Sheaves 35 3.1 Presheaves and Sheaves . 35 3.1.1 What is a Presheaf? . 35 3.1.2 A Remark about Products and Presheaf . 36 3.1.3 What is a Sheaf? . 36 3.1.4 Examples . 38 3.2 Manifolds as Locally Ringed Spaces . 39 3.2.1 What Are Manifolds? . 39 3.2.2 Examples and Exercise . 41 3.3 Stalks and Sheafification . 45 3.3.1 Stalks . 45 3.3.2 The Process of Sheafification of a Presheaf . 46 3.4 The Functors ƒ* and ƒ. 47 3.4.1 The Adjunction Formula . 48 3.4.2 Extensions and Restrictions . 49 3.5 Constructions of Sheaves . 49 Cohomology of Sheaves 51 4.1 Examples . 51 4.1.1 Sheaves on Riemann surfaces . 51 4.1.2 Cohomology of the Circle . 54 4.2 The Derived Functor . 55 4.2.1 Injective Sheaves and Derived Functors . 55 4.2.2 A Direct Definition of Я1 . 56 4.3 Fiber Bundles and Non Abelian H1 . 59 4.3.1 Fibrations . 59 Fibre Bundle . 59 Vector Bundles . 60 4.3.2 Non- Abelian Я1 . 61 4.3.3 The Reduction of the Structure Group . 62 Orientation . 62 Local Systems . 63 Isomorphism Classes of Local Systems . 64 Principal G-bundels . 64 4.4 Fundamental Properties of the Cohomology of Sheaves . 65 4.4.1 Introduction . 65 4.4.2 The Derived Functor to ƒ . 66 4.4.3 Functorial Properties of the Cohomology . 67 4.4.4 Paracompact Spaces . 69 4.4.5 Applications . 75 Cohomology of Spheres . 75 Orientations . 76 Compact Oriented Surfaces . 77 4.5 Cech Cohomology of Sheaves . 77 4.5.1 The Čech-Complex . 77 4.5.2 The Čech Resolution of a Sheaf . 80 4.6 Spectral Sequences . 82 4.6.1 Introduction . 82 4.6.2 The Vertical Filtration . 87 4.6.3 The Horizontal Filtration . 93 Two Special Cases . 94 Applications of Spectral Sequences . 95 4.6.4 The Derived Category . 97 The Composition Rule . 100 Exact Triangles . 101 4.6.5 The Spectral Sequence of a Fibration . 102 Sphere Bundles an Euler Characteristic . 103 4.6.6 Čech Complexes and the Spectral Sequence . 104 A Criterion for Degeneration . 106 An Application to Product Spaces . 108 4.6.7 The Cup Product . 110 4.6.8 Example: Cup Product for the Comology of Tori . 114 A Connection to the Cohomology of Groups . 115 4.6.9 An Excursion into Homotopy Theory . 116 4.7 Cohomology with Compact Supports .-. 119 4.7.1 The Definition . 119 4.7.2 An Example for Cohomology with Compact Supports . 120 The Cohomology with Compact Supports for Open Balls . 120 Formulae for Cup Products . 122 4.7.3 The Fundamental Class . 124 4.8 Cohomology of Manifolds . 125 4.8.1 Local Systems . 125 4.8.2 Čech Resolutions of Local Systems . 126 4.8.3 Čech Coresolution of Local Systems . 128 4.8.4 Poincaré Duality . 131 4.8.5 The Cohomology in Top Degree and the Homology . 137 4.8.6 Some Remarks on Singular Homology . 139 4.8.7 Cohomology with Compact Support and Embeddings . 140 4.8.8 The Fundamental Class of a Submanifold . 142 4.8.9 Cup Product and Intersections . 143 4.8.10 Compact oriented Surfaces . 145 4.8.11 The Cohomology Ring of Pn(C) . 146 4.9 The Lefschetz Fixed Point Formula . 146 4.9.1 The Euler Characteristic of Manifolds . 148 4.10 The de Rham and the Dolbeault Isomorphism . 149 4.10.1 The Cohomology of Flat Bundles on Real Manifolds . 149 The Product Structure on the de Rham Cohomology . 152 The de Rham Isomorphism and the fundamental class . 153 4.10.2 Cohomology of Holomorphic Bundles on Complex Manifolds . . . 155 The Tangent Bundle . 155 The Bundle tl™ . 157 4.10.3 Chern Classes . 159 The Line Bundles O¥n{c)(k) . 162 4.11 Hodge Theory . 163 4.11.1 Hodge Theory on Real Manifolds . 163 4.11.2 Hodge Theory on Complex Manifolds . 168 Some Linear Algebra . 168 Kahler Manifolds and their Cohomology . 171 The Cohomology of Holomorphic Vector Bundles . 174 Serre Duality . 175 4.11.3 Hodge Theory on Tori . 176 5 Compact Riemann surfaces and Abelian Varieties 179 5.1 Compact Riemann Surfaces . 179 5.1.1 Introduction . 179 5.1.2 The Hodge Structure on ff1 (S1,C) . 180 5.1.3 Cohomology of Holomorphic Bundles . 185 5.1.4 The Theorem of Riemann- Roch. 191 On the Picard Group . 191 Exercises . 192 The Theorem of Riemann-Roch . 193 5.1.5 The Algebraic Duality Pairing . 194 5.1.6 Riemann Surfaces of Low Genus . 196 5.1.7 The Algebraicity of Riemann Surfaces . 197 From a Riemann Surface to Function Fields . 197 The reconstruction of S from К . 202 Connection to Algebraic Geometry . 209 Elliptic Curves . 211 5.1.8 Géométrie Analytique et Géométrie Algébrique - GAGA. 212 5.1.9 Comparison of Two Pairings . 215 5.1.10 The Jacobian of a Compact Riemann Surface . 217 5.1.11 The Classical Version of Abel's Theorem . 218 5.1.12 Riemann Period Relations . 222 5.2 Line Bundles on Complex Tori . 223 5.2.1 Construction of Line Bundles . 223 The Poincaré Bundle . 229 Universality of AT . 230 5.2.2 Homomorphisms Between Complex Tori . 232 The Neron Severi group and Нот(Л,Лу) . 234 The construction of Φ starting from a line bundle . 235 5.2.3 The Self Duality of the Jacobian . 236 5.2.4 Ample Line Bundles and the Algebraicity of the Jacobian . 237 The Kodaira Embedding Theorem . 237 The Spaces of Sections . 239 5.2.5 The Siegel Upper Half Space . 240 5.2.6 Riemann-Theta Functions . 243 5.2.7 Projective embeddings of abelian varieties . 248 5.2.8 Degeneration of Abelian Varieties . 250 The Case of Genus 1. 251 The Algebraic Approach . 261 5.3 Towards the Algebraic Theory . 263 5.3.1 Introduction . 263 The Algebraic Definition of the Neron-Severi Group . 264 The Algebraic Definition of the Intersection Numbers . 264 The Study of some Special Neron-Severi groups . 266 Xl 5.3.2 The Structure of End( J) .270 The Rosati Involution .270 A Trace Formula .272 The Fundamental Class [S] of S under the Abel Map .275 5.3.3 The Ring of Correspondences . 276 5.3.4 An Algebraic Substitute for the Cohomology .277 Bibliography 281 Index 284 GIINTER HARDER LECTURES ON ALGEBRAIC GEOMETRY I SHEAVES, COHOMOLOGY OF SHEAVES, AND APPLICATIONS TO RIEMANN SURFACES VIEWEG CONTENTS PREFACE V INTRODUCTION XIII 1 CATEGORIES, PRODUCTS, PROJECTIVE AND INDUCTIVE LIMITS 1 1.1 THE NOTION OF A CATEGORY AND EXAMPLES 1 1.2 FUNCTORS 3 1.3 PRODUCTS, PROJECTIVE LIMITS AND DIRECT LIMITS IN A CATEGORY 4 1.3.1 THE PROJECTIVE LIMIT 4 1.3.2 THE YONEDA LEMMA 6 1.3.3 EXAMPLES 6 1.3.4 REPRESENTABLE FUNCTORS 8 1.3.5 DIRECT LIMITS 9 1.4 EXERCISES 10 2 BASIC CONCEPTS OF HOMOLOGICAL ALGEBRA 11 2.1 THE CATEGORY MODR OF F-MODULES 11 2.2 MORE FUNCTORS 13 2.2.1 INVARIANTS, COINVARIANTS AND EXACTNESS 13 2.2.2 THE FIRST COHOMOLOGY GROUP 15 2.2.3 SOME NOTATION 16 2.2.4 EXERCISES 17 2.3 THE DERIVED FUNCTORS 19 2.3.1 THE SIMPLE PRINCIPLE 20 2.3.2 FUNCTORIALITY 22 2.3.3 OTHER RESOLUTIONS 24 2.3.4 INJECTIVE RESOLUTIONS OF SHORT EXACT SEQUENCES 24 A FUNDAMENTAL REMARK 26 THE COHOMOLOGY AND THE LONG EXACT SEQUENCE 27 THE HOMOLOGY OF GROUPS 27 2.4 THE FUNCTORS EXT AND TOR 28 2.4.1 THE FUNCTOR EXT 28 2.4.2 THE DERIVED FUNCTOR FOR THE TENSOR PRODUCT 30 2.4.3 EXERCISE 32 3 SHEAVES 35 3.1 PRESHEAVES AND SHEAVES 35 3.1.1 WHAT IS A PRESHEAF? 35 3.1.2 A REMARK ABOUT PRODUCTS AND PRESHEAF 36 3.1.3 WHAT IS A SHEAF? 36 3.1.4 EXAMPLES 38 3.2 MANIFOLDS AS LOCALLY RINGED SPACES 39 I CONTENTS 3.2.1 WHAT ARE MANIFOLDS? 39 3.2.2 EXAMPLES AND EXERCISE 41 3.3 STALKS AND SHEAFIFICATION 45 3.3.1 STALKS 45 3.3.2 THE PROCESS OF SHEAFIFICATION OF A PRESHEAF 46 3.4 THE FUNCTORS /, AND /* 47 3.4.1 THE ADJUNCTION FORMULA 48 3.4.2 EXTENSIONS AND RESTRICTIONS 49 3.5 CONSTRUCTIONS OF SHEAVES 49 COHOMOLOGY OF SHEAVES 51 4.1 EXAMPLES 51 4.1.1 SHEAVES ON RIEMANN SURFACES 51 4.1.2 COHOMOLOGY OF THE CIRCLE 54 4.2 THE DERIVED FUNCTOR 55 4.2.1 INJECTIVE SHEAVES AND DERIVED FUNCTORS 55 4.2.2 A DIRECT DEFINITION OF H 1 56 4.3 FIBER BUNDLES AND NON ABELIAN H 1 59 4.3.1 FIBRATIONS 59 FIBRE BUNDLE 59 VECTOR BUNDLES 60 4.3.2 NON-ABELIAN H 1 61 4.3.3 THE REDUCTION OF THE STRUCTURE GROUP 62 ORIENTATION 62 LOCAL SYSTEMS 63 ISOMORPHISM CLASSES OF LOCAL SYSTEMS 64 PRINCIPAL G-BUNDELS 64 4.4 FUNDAMENTAL PROPERTIES OF THE COHOMOLOGY OF SHEAVES 65 4.4.1 INTRODUCTION 65 4.4.2 THE DERIVED FUNCTOR TO /, 66 4.4.3 FUNCTORIAL PROPERTIES OF THE COHOMOLOGY 67 4.4.4 PARACOMPACT SPACES 69 4.4.5 APPLICATIONS 75 COHOMOLOGY OF SPHERES 75 ORIENTATIONS 76 COMPACT ORIENTED SURFACES 77 4.5 CECH COHOMOLOGY OF SHEAVES 77 4.5.1 THE CECH-COMPLEX 77 4.5.2 THE CECH RESOLUTION OF A SHEAF 80 4.6 SPECTRAL SEQUENCES 82 4.6.1 INTRODUCTION 82 4.6.2 THE VERTICAL FILTRATION 87 4.6.3 THE HORIZONTAL FILTRATION 93 TWO SPECIAL CASES 94 APPLICATIONS OF SPECTRAL SEQUENCES 95 4.6.4 THE DERIVED CATEGORY 97 THE COMPOSITION RULE 100 EXACT TRIANGLES 101 4.6.5 THE SPECTRAL SEQUENCE OF A FIBRATION 102 SPHERE BUNDLES AN EULER CHARACTERISTIC 103 4.6.6 CECH COMPLEXES AND THE SPECTRAL SEQUENCE 104 A CRITERION FOR DEGENERATION 106 AN APPLICATION TO PRODUCT SPACES 108 4.6.7 THE CUP PRODUCT 110 4.6.8 EXAMPLE: CUP PRODUCT FOR THE COMOLOGY OF TORI 114 A CONNECTION TO THE COHOMOLOGY OF GROUPS 115 4.6.9 AN EXCURSION INTO HOMOTOPY THEORY 116 4.7 COHOMOLOGY WITH COMPACT SUPPORTS 119 4.7.1 THE DEFINITION 119 4.7.2 AN EXAMPLE FOR COHOMOLOGY WITH COMPACT SUPPORTS 120 THE COHOMOLOGY WITH COMPACT SUPPORTS FOR OPEN BALLS 120 FORMULAE FOR CUP PRODUCTS 122 4.7.3 THE FUNDAMENTAL CLASS 124 4.8 COHOMOLOGY OF MANIFOLDS 125 4.8.1 LOCAL SYSTEMS 125 4.8.2 CECH RESOLUTIONS OF LOCAL SYSTEMS 126 4.8.3 CECH CORESOLUTION OF LOCAL SYSTEMS 128 4.8.4 POINCARE DUALITY 131 4.8.5 THE COHOMOLOGY IN TOP DEGREE AND THE HOMOLOGY 137 4.8.6 SOME REMARKS ON SINGULAR HOMOLOGY 139 4.8.7 COHOMOLOGY WITH COMPACT SUPPORT AND EMBEDDINGS 140 4.8.8 THE FUNDAMENTAL CLASS OF A SUBMANIFOLD 142 4.8.9 CUP PRODUCT AND INTERSECTIONS 143 4.8.10 COMPACT ORIENTED SURFACES 145 4.8.11 THE COHOMOLOGY RING OF P N (C) 146 4.9 THE LEFSCHETZ FIXED POINT FORMULA 146 4.9.1 THE EULER CHARACTERISTIC OF MANIFOLDS 148 4.10 THE DE RHAM AND THE DOLBEAULT ISOMORPHISM 149 4.10.1 THE COHOMOLOGY OF FLAT BUNDLES ON REAL MANIFOLDS 149 THE PRODUCT STRUCTURE ON THE DE RHAM COHOMOLOGY 152 THE DE RHAM ISOMORPHISM AND THE FUNDAMENTAL CLASS 153 4.10.2 COHOMOLOGY OF HOLOMORPHIC BUNDLES ON COMPLEX MANIFOLDS . . . 155 THE TANGENT BUNDLE 155 THE BUNDLE &$ 157 4.10.3 CHERN CLASSES 159 THE LINE BUNDLES O PN(C ){K) 162 4.11 HODGE THEORY 163 4.11.1 HODGE THEORY ON REAL MANIFOLDS 163 4.11.2 HODGE THEORY ON COMPLEX MANIFOLDS 168 SOME LINEAR ALGEBRA 168 KAHLER MANIFOLDS AND THEIR COHOMOLOGY 171 THE COHOMOLOGY OF HOLOMORPHIC VECTOR BUNDLES 174 SERRE DUALITY 175 4.11.3 HODGE THEORY ON TORI 176 CONTENTS COMPACT RIEMANN SURFACES AND ABELIAN VARIETIES 179 5.1 COMPACT RIEMANN SURFACES 179 5.1.1 INTRODUCTION 179 5.1.2 THE HODGE STRUCTURE ON IF 1 (5,0) 180 5.1.3 COHOMOLOGY OF HOLOMORPHIC BUNDLES 185 5.1.4 THE THEOREM OF RIEMANN-ROCH 191 ON THE PICARD GROUP 191 EXERCISES 192 THE THEOREM OF RIEMANN-ROCH 193 5.1.5 THE ALGEBRAIC DUALITY PAIRING 194 5.1.6 RIEMANN SURFACES OF LOW GENUS 196 5.1.7 THE ALGEBRAICITY OF RIEMANN SURFACES 197 FROM A RIEMANN SURFACE TO FUNCTION FIELDS 197 THE RECONSTRUCTION OF S FROM K 202 CONNECTION TO ALGEBRAIC GEOMETRY 209 ELLIPTIC CURVES 211 5.1.8 GEOMETRIE ANALYTIQUE ET GEOMETRIE ALGEBRIQUE - GAGA 212 5.1.9 COMPARISON OF TWO PAIRINGS 215 5.1.10 THE JACOBIAN OF A COMPACT RIEMANN SURFACE 217 5.1.11 THE CLASSICAL VERSION OF ABEL'S THEOREM 218 5.1.12 RIEMANN PERIOD RELATIONS 222 5.2 LINE BUNDLES ON COMPLEX TORI 223 5.2.1 CONSTRUCTION OF LINE BUNDLES 223 THE POINCARE BUNDLE 229 UNIVERSALITY OF M 230 5.2.2 HOMOMORPHISMS BETWEEN COMPLEX TORI 232 THE NERON SEVERI GROUP AND HOM(,4,.4 V ) 234 THE CONSTRUCTION OF \F STARTING FROM A LINE BUNDLE 235 5.2.3 THE SELF DUALITY OF THE JACOBIAN 236 5.2.4 AMPLE LINE BUNDLES AND THE ALGEBRAICITY OF THE JACOBIAN 237 THE KODAIRA EMBEDDING THEOREM 237 THE SPACES OF SECTIONS 239 5.2.5 THE SIEGEL UPPER HALF SPACE 240 5.2.6 RIEMANN-THETA FUNCTIONS 243 5.2.7 PROJECTIVE EMBEDDINGS OF ABELIAN VARIETIES 248 5.2.8 DEGENERATION OF ABELIAN VARIETIES 250 THE CASE OF GENUS 1 251 THE ALGEBRAIC APPROACH 261 5.3 TOWARDS THE ALGEBRAIC THEORY 263 5.3.1 INTRODUCTION 263 THE ALGEBRAIC DEFINITION OF THE NERON-SEVERI GROUP 264 THE ALGEBRAIC DEFINITION OF THE INTERSECTION NUMBERS 264 THE STUDY OF SOME SPECIAL NERON-SEVERI GROUPS 266 5.3.2 THE STRUCTURE OF END( J) 270 THE ROSATI INVOLUTION 270 A TRACE FORMULA 272 THE FUNDAMENTAL CLASS [S] OF S UNDER THE ABEL MAP 275 5.3.3 THE RING OF CORRESPONDENCES 276 5.3.4 AN ALGEBRAIC SUBSTITUTE FOR THE COHOMOLOGY 277 BIBLIOGRAPHY 281 INDEX 284
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spelling	Harder, Günter Verfasser aut Lectures on algebraic geometry 1 Sheaves, cohomology of sheaves, and applications to Riemann surfaces Günter Harder 1. ed. Wiesbaden Vieweg 2008 XIV, 290 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Aspects of mathematics : E 35 Aspects of mathematics : E ... (DE-604)BV023059274 1 Aspects of mathematics E ; 35 (DE-604)BV000018737 35 Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016262537&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016262537&sequence=000003&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Harder, Günter Lectures on algebraic geometry Aspects of mathematics
title	Lectures on algebraic geometry
title_auth	Lectures on algebraic geometry
title_exact_search	Lectures on algebraic geometry
title_exact_search_txtP	Lectures on algebraic geometry
title_full	Lectures on algebraic geometry 1 Sheaves, cohomology of sheaves, and applications to Riemann surfaces Günter Harder
title_fullStr	Lectures on algebraic geometry 1 Sheaves, cohomology of sheaves, and applications to Riemann surfaces Günter Harder
title_full_unstemmed	Lectures on algebraic geometry 1 Sheaves, cohomology of sheaves, and applications to Riemann surfaces Günter Harder
title_short	Lectures on algebraic geometry
title_sort	lectures on algebraic geometry sheaves cohomology of sheaves and applications to riemann surfaces
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