Lectures on algebraic geometry: 1 Sheaves, cohomology of sheaves, and applications to Riemann surfaces
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| Format: | Book |
| Language: | English |
| Published: |
Wiesbaden
Vieweg
2008
|
| Edition: | 1. ed. |
| Series: | Aspects of mathematics
E ; 35 |
| Online Access: | Inhaltsverzeichnis Inhaltsverzeichnis |
| Physical Description: | XIV, 290 S. graph. Darst. |
| ISBN: | 9783528031367 |
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| 020 | |a 9783528031367 |9 978-3-528-03136-7 | ||
| 035 | |a (OCoLC)313567622 | ||
| 035 | |a (DE-599)BVBBV023059278 | ||
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| 100 | 1 | |a Harder, Günter |d 1938-2025 |e Verfasser |0 (DE-588)1011622483 |4 aut | |
| 245 | 1 | 0 | |a Lectures on algebraic geometry |n 1 |p Sheaves, cohomology of sheaves, and applications to Riemann surfaces |c Günter Harder |
| 250 | |a 1. ed. | ||
| 264 | 1 | |c 2008 | |
| 264 | 1 | |a Wiesbaden |b Vieweg | |
| 300 | |a XIV, 290 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Aspects of mathematics : E |v 35 | |
| 773 | 0 | 8 | |w (DE-604)BV023059274 |g 1 |
| 830 | 0 | |a Aspects of mathematics |v E ; 35 |w (DE-604)BV000018737 |9 35 | |
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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 |
| any_adam_object | |
| any_adam_object_boolean | 1 |
| author | Harder, Günter 1938-2025 |
| author_GND | (DE-588)1011622483 |
| author_facet | Harder, Günter 1938-2025 |
| author_role | aut |
| author_sort | Harder, Günter 1938-2025 |
| author_variant | g h gh |
| building | Verbundindex |
| bvnumber | BV023059278 |
| classification_rvk | SK 240 |
| ctrlnum | (OCoLC)313567622 (DE-599)BVBBV023059278 |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | 1. ed. |
| format | Book |
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| id | DE-604.BV023059278 |
| illustrated | Illustrated |
| index_date | 2024-07-02T19:28:06Z |
| indexdate | 2025-06-17T20:00:14Z |
| institution | BVB |
| isbn | 9783528031367 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016262537 |
| oclc_num | 313567622 |
| open_access_boolean | |
| owner | DE-703 DE-739 DE-824 DE-20 DE-29T DE-19 DE-BY-UBM DE-634 DE-83 DE-11 |
| owner_facet | DE-703 DE-739 DE-824 DE-20 DE-29T DE-19 DE-BY-UBM DE-634 DE-83 DE-11 |
| physical | XIV, 290 S. graph. Darst. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Vieweg |
| record_format | marc |
| series | Aspects of mathematics |
| series2 | Aspects of mathematics : E |
| spelling | Harder, Günter 1938-2025 Verfasser (DE-588)1011622483 aut Lectures on algebraic geometry 1 Sheaves, cohomology of sheaves, and applications to Riemann surfaces Günter Harder 1. ed. 2008 Wiesbaden Vieweg XIV, 290 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Aspects of mathematics : E 35 (DE-604)BV023059274 1 Aspects of mathematics E ; 35 (DE-604)BV000018737 35 Digitalisierung UB Passau https://bvbr.bib-bvb.de:443/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 https://bvbr.bib-bvb.de:443/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 1938-2025 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 |
| url | https://bvbr.bib-bvb.de:443/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016262537&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA https://bvbr.bib-bvb.de:443/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016262537&sequence=000003&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV023059274 (DE-604)BV000018737 |
| work_keys_str_mv | AT hardergunter lecturesonalgebraicgeometry1 |
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