Positivity in algebraic geometry:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin ; Heidelberg ; New York
Springer
[2004]
|
| Series: | Ergebnisse der Mathematik und ihrer Grenzgebiete : 3. Folge
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
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MARC
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| 100 | 1 | |a Lazarsfeld, Robert |d 1953- |e Verfasser |0 (DE-588)102154279 |4 aut | |
| 245 | 1 | 0 | |a Positivity in algebraic geometry |c Robert Lazarsfeld |
| 264 | 1 | |a Berlin ; Heidelberg ; New York |b Springer |c [2004] | |
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| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Ergebnisse der Mathematik und ihrer Grenzgebiete : 3. Folge | |
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| adam_text | CONTENTS
NOTATION AND CONVENTIONS
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
1
PART ONE: AMPLE LINE BUNDLES AND LINEAR SERIES
INTRODUCTION TO PART ONE
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. .
5
1 AMPLE AND NEF LINE BUNDLES
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.1 PRELIMINARIES: DIVISORS, LINE BUNDLES, AND LINEAR SERIES . . . . . .
7
1.1.A DIVISORS AND LINE BUNDLES . . . . . . . . . . . . . . . . . . . .
. . . . . 8
1.1.B LINEAR SERIES . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 12
1.1.C INTERSECTION NUMBERS AND NUMERICAL EQUIVALENCE . . . . . 15
1.1.D RIEMANN*ROCH . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 20
1.2 THE CLASSICAL THEORY . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 24
1.2.A COHOMOLOGICAL PROPERTIES . . . . . . . . . . . . . . . . . . . . .
. . . . . 25
1.2.B NUMERICAL PROPERTIES . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 33
1.2.C METRIC CHARACTERIZATIONS OF AMPLITUDE . . . . . . . . . . . . . .
39
1.3
Q
-DIVISORS AND
R
-DIVISORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 44
1.3.A DEFINITIONS FOR
Q
-DIVISORS . . . . . . . . . . . . . . . . . . . . . . . . . . 44
1.3.B
R
-DIVISORS AND THEIR AMPLITUDE . . . . . . . . . . . . . . . . . . . . 48
1.4 NEF LINE BUNDLES AND DIVISORS . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 50
1.4.A DEFINITIONS AND FORMAL PROPERTIES . . . . . . . . . . . . . . . .
. . . 51
1.4.B KLEIMAN S THEOREM . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 53
1.4.C CONES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 59
1.4.D FUJITA S VANISHING THEOREM . . . . . . . . . . . . . . . . . . . .
. . . . 65
1.5 EXAMPLES AND COMPLEMENTS . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 70
1.5.A RULED SURFACES . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 70
1.5.B PRODUCTS OF CURVES . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 73
1.5.C ABELIAN VARIETIES . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 79
1.5.D OTHER VARIETIES . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 80
1.5.E LOCAL STRUCTURE OF THE NEF CONE . . . . . . . . . . . . . . . . .
. . . 82
XII CONTENTS
1.5.F THE CONE THEOREM . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 86
1.6 INEQUALITIES . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 88
1.6.A GLOBAL RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 88
1.6.B MIXED MULTIPLICITIES . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 91
1.7 AMPLITUDE FOR A MAPPING . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 94
1.8 CASTELNUOVO*MUMFORD REGULARITY. . . . . . . . . . . . . . . . . . .
. . . . . . . 98
1.8.A DEFINITIONS, FORMAL PROPERTIES, AND VARIANTS . . . . . . . . . .
99
1.8.B REGULARITY AND COMPLEXITY . . . . . . . . . . . . . . . . . . . .
. . . . . 107
1.8.C REGULARITY BOUNDS . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 110
1.8.D SYZYGIES OF ALGEBRAIC VARIETIES . . . . . . . . . . . . . . . . .
. . . . . 115
NOTES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 119
2 LINEAR SERIES
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . .
121
2.1 ASYMPTOTIC THEORY . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 121
2.1.A BASIC DEFINITIONS . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 122
2.1.B SEMIAMPLE LINE BUNDLES . . . . . . . . . . . . . . . . . . . . . .
. . . . . 128
2.1.C IITAKA FIBRATION . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 133
2.2 BIG LINE BUNDLES AND DIVISORS . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 139
2.2.A BASIC PROPERTIES OF BIG DIVISORS . . . . . . . . . . . . . . . . .
. . . 139
2.2.B PSEUDOEECTIVE AND BIG CONES . . . . . . . . . . . . . . . . . . .
. . .
145
2.2.C VOLUME OF A BIG DIVISOR . . . . . . . . . . . . . . . . . . . . .
. . . . . . 148
2.3 EXAMPLES AND COMPLEMENTS . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 157
2.3.A ZARISKI S CONSTRUCTION . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 158
2.3.B CUTKOSKY S CONSTRUCTION . . . . . . . . . . . . . . . . . . . . .
. . . . . . 159
2.3.C BASE LOCI OF NEF AND BIG LINEAR SERIES . . . . . . . . . . . . . .
164
2.3.D THE THEOREM OF CAMPANA AND PETERNELL . . . . . . . . . . . . . 166
2.3.E ZARISKI DECOMPOSITIONS . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 167
2.4 GRADED LINEAR SERIES AND FAMILIES OF IDEALS . . . . . . . . . . . .
. . . . . 172
2.4.A GRADED LINEAR SERIES . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 172
2.4.B GRADED FAMILIES OF IDEALS . . . . . . . . . . . . . . . . . . . .
. . . . . . 176
NOTES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 183
3 GEOMETRIC MANIFESTATIONS OF POSITIVITY
. . . . . . . . . . . . . . . . . . . .
185
3.1 THE LEFSCHETZ THEOREMS . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 185
3.1.A TOPOLOGY OF ANE VARIETIES . . . . . . . . . . . . . . . . . . . .
. . . .
186
3.1.B THE THEOREM ON HYPERPLANE SECTIONS . . . . . . . . . . . . . . .
192
3.1.C HARD LEFSCHETZ THEOREM . . . . . . . . . . . . . . . . . . . . . .
. . . . . 199
3.2 PROJECTIVE SUBVARIETIES OF SMALL CODIMENSION . . . . . . . . . . . .
. . . 201
3.2.A BARTH S THEOREM . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 201
3.2.B HARTSHORNE S CONJECTURES . . . . . . . . . . . . . . . . . . . . .
. . . . . 204
3.3 CONNECTEDNESS THEOREMS . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 207
3.3.A BERTINI THEOREMS . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 207
3.3.B THEOREM OF FULTON AND HANSEN . . . . . . . . . . . . . . . . . . .
. . 210
3.3.C GROTHENDIECK S CONNECTEDNESS THEOREM . . . . . . . . . . . . . 212
3.4 APPLICATIONS OF THE FULTON*HANSEN THEOREM . . . . . . . . . . . . .
. . . 213
3.4.A SINGULARITIES OF MAPPINGS . . . . . . . . . . . . . . . . . . . .
. . . . . . 214
CONTENTS XIII
3.4.B ZAK S THEOREMS . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 219
3.4.C ZARISKI S PROBLEM . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 227
3.5 VARIANTS AND EXTENSIONS . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 231
3.5.A HOMOGENEOUS VARIETIES . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 231
3.5.B HIGHER CONNECTIVITY . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 233
NOTES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 237
4 VANISHING THEOREMS
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . .
239
4.1 PRELIMINARIES . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 240
4.1.A NORMAL CROSSINGS AND RESOLUTIONS OF SINGULARITIES . . . . . 240
4.1.B COVERING LEMMAS . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 242
4.2 KODAIRA AND NAKANO VANISHING THEOREMS . . . . . . . . . . . . . . .
. . . 248
4.3 VANISHING FOR BIG AND NEF LINE BUNDLES . . . . . . . . . . . . . . .
. . . . . 252
4.3.A STATEMENT AND PROOF OF THE THEOREM . . . . . . . . . . . . . . . .
252
4.3.B SOME APPLICATIONS . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 257
4.4 GENERIC VANISHING THEOREM . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 261
NOTES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 267
5 LOCAL POSITIVITY
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . .
269
5.1 SESHADRI CONSTANTS . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 269
5.2 LOWER BOUNDS . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 278
5.2.A BACKGROUND AND STATEMENTS . . . . . . . . . . . . . . . . . . . .
. . . . 278
5.2.B MULTIPLICITIES OF DIVISORS IN FAMILIES . . . . . . . . . . . . . .
. . . 282
5.2.C PROOF OF THEOREM 5.2.5 . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 286
5.3 ABELIAN VARIETIES . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 290
5.3.A PERIOD LENGTHS AND SESHADRI CONSTANTS . . . . . . . . . . . . . .
290
5.3.B PROOF OF THEOREM 5.3.6 . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 297
5.3.C COMPLEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 301
5.4 LOCAL POSITIVITY ALONG AN IDEAL SHEAF . . . . . . . . . . . . . . .
. . . . . . . 303
5.4.A DEFINITION AND FORMAL PROPERTIES OF THE S-INVARIANT . . . . 303
5.4.B COMPLEXITY BOUNDS . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 308
NOTES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 312
APPENDICES
A PROJECTIVE BUNDLES
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . .
315
B COHOMOLOGY AND COMPLEXES
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
317
B.1 COHOMOLOGY . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 317
B.2 COMPLEXES . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 320
REFERENCES
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
325
GLOSSARY OF NOTATION
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . .
359
INDEX
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . .
365
CONTENTS OF VOLUME II
NOTATION AND CONVENTIONS
II
: 1
PART TWO: POSITIVITY FOR VECTOR BUNDLES
INTRODUCTION TO PART TWO
II
: 5
6 AMPLE AND NEF VECTOR BUNDLES
II
: 7
6.1 CLASSICAL THEORY
II
: 7
6.1.A DEFINITION AND FIRST PROPERTIES
II
: 8
6.1.B COHOMOLOGICAL PROPERTIES
II
: 11
6.1.C CRITERIA FOR AMPLITUDE
II
: 15
6.1.D METRIC APPROACHES TO POSITIVITY OF VECTOR BUNDLES
II
: 18
6.2
Q
-TWISTED AND NEF BUNDLES
II
: 20
6.2.A TWISTS BY
Q
-DIVISORS
II
: 20
6.2.B NEF BUNDLES
II
: 24
6.3 EXAMPLES AND CONSTRUCTIONS
II
: 27
6.3.A NORMAL AND TANGENT BUNDLES
II
: 27
6.3.B AMPLE COTANGENT BUNDLES AND HYPERBOLICITY
II
: 36
6.3.C PICARD BUNDLES
II
: 44
6.3.D THE BUNDLE ASSOCIATED TO A BRANCHED COVERING
II
: 47
6.3.E DIRECT IMAGES OF CANONICAL BUNDLES
II
: 51
6.3.F SOME CONSTRUCTIONS OF POSITIVE VECTOR BUNDLES
II
: 53
6.4 AMPLE VECTOR BUNDLES ON CURVES
II
: 56
6.4.A REVIEW OF SEMISTABILITY
II
: 57
6.4.B SEMISTABILITY AND AMPLITUDE
II
: 60
NOTES
II
: 64
7 GEOMETRIC PROPERTIES OF AMPLE BUNDLES
II
: 65
7.1 TOPOLOGY
II
: 65
7.1.A SOMMESE S THEOREM
II
: 65
XVI CONTENTS OF VOLUME II
7.1.B THEOREM OF BLOCH AND GIESEKER
II
: 68
7.1.C A BARTH-TYPE THEOREM FOR BRANCHED COVERINGS
II
: 71
7.2 DEGENERACY LOCI
II
: 74
7.2.A STATEMENTS AND FIRST EXAMPLES
II
: 74
7.2.B PROOF OF CONNECTEDNESS OF DEGENERACY LOCI
II
: 78
7.2.C SOME APPLICATIONS
II
: 82
7.2.D VARIANTS AND EXTENSIONS
II
: 87
7.3 VANISHING THEOREMS
II
: 89
7.3.A VANISHING THEOREMS OF GRITHS AND LE POTIER
II
: 89
7.3.B GENERALIZATIONS
II
: 95
NOTES
II
: 98
8 NUMERICAL PROPERTIES OF AMPLE BUNDLES
II
: 101
8.1 PRELIMINARIES FROM INTERSECTION THEORY
II
: 101
8.1.A CHERN CLASSES FOR
Q
-TWISTED BUNDLES
II
: 102
8.1.B CONE CLASSES
II
: 104
8.1.C CONE CLASSES FOR
Q
-TWISTS
II
: 110
8.2 POSITIVITY THEOREMS
II
: 111
8.2.A POSITIVITY OF CHERN CLASSES
II
: 111
8.2.B POSITIVITY OF CONE CLASSES
II
: 114
8.3 POSITIVE POLYNOMIALS FOR AMPLE BUNDLES
II
: 117
8.4 SOME APPLICATIONS
II
: 125
8.4.A POSITIVITY OF INTERSECTION PRODUCTS
II
: 125
8.4.B NON-EMPTINESS OF DEGENERACY LOCI
II
: 127
8.4.C SINGULARITIES OF HYPERSURFACES ALONG A CURVE
II
: 129
NOTES
II
: 132
PART THREE: MULTIPLIER IDEALS AND THEIR APPLICATIONS
INTRODUCTION TO PART THREE
II
: 135
9 MULTIPLIER IDEAL SHEAVES
II
: 139
9.1 PRELIMINARIES
II
: 140
9.1.A
Q
-DIVISORS
II
: 140
9.1.B NORMAL CROSSING DIVISORS AND LOG RESOLUTIONS
II
: 142
9.1.C THE KAWAMATA*VIEHWEG VANISHING THEOREM
II
: 147
9.2 DEFINITION AND FIRST PROPERTIES
II
: 151
9.2.A DEFINITION OF MULTIPLIER IDEALS
II
: 152
9.2.B FIRST PROPERTIES
II
: 158
9.3 EXAMPLES AND COMPLEMENTS
II
: 162
9.3.A MULTIPLIER IDEALS AND MULTIPLICITY
II
: 162
9.3.B INVARIANTS ARISING FROM MULTIPLIER IDEALS
II
: 165
9.3.C MONOMIAL IDEALS
II
: 170
9.3.D ANALYTIC CONSTRUCTION OF MULTIPLIER IDEALS
II
: 176
CONTENTS OF VOLUME II XVII
9.3.E ADJOINT IDEALS
II
: 179
9.3.F MULTIPLIER AND JACOBIAN IDEALS
II
: 181
9.3.G MULTIPLIER IDEALS ON SINGULAR VARIETIES
II
: 182
9.4 VANISHING THEOREMS FOR MULTIPLIER IDEALS
II
: 185
9.4.A LOCAL VANISHING FOR MULTIPLIER IDEALS
II
: 186
9.4.B THE NADEL VANISHING THEOREM
II
: 188
9.4.C VANISHING ON SINGULAR VARIETIES
II
: 191
9.4.D NADEL S THEOREM IN THE ANALYTIC SETTING
II
: 192
9.4.E NON-VANISHING AND GLOBAL GENERATION
II
: 193
9.5 GEOMETRIC PROPERTIES OF MULTIPLIER IDEALS
II
: 195
9.5.A RESTRICTIONS OF MULTIPLIER IDEALS
II
: 195
9.5.B SUBADDITIVITY
II
: 201
9.5.C THE SUMMATION THEOREM
II
: 204
9.5.D MULTIPLIER IDEALS IN FAMILIES
II
: 210
9.5.E COVERINGS
II
: 213
9.6 SKODA S THEOREM
II
: 216
9.6.A INTEGRAL CLOSURE OF IDEALS
II
: 216
9.6.B SKODA S THEOREM: STATEMENTS
II
: 221
9.6.C SKODA S THEOREM: PROOFS
II
: 226
9.6.D VARIANTS
II
: 228
NOTES
II
: 230
10 SOME APPLICATIONS OF MULTIPLIER IDEALS
II
: 233
10.1 SINGULARITIES
II
: 233
10.1.A SINGULARITIES OF PROJECTIVE HYPERSURFACES
II
: 233
10.1.B SINGULARITIES OF THETA DIVISORS
II
: 235
10.1.C A CRITERION FOR SEPARATION OF JETS OF ADJOINT SERIES
II
: 238
10.2 MATSUSAKA S THEOREM
II
: 239
10.3 NAKAMAYE S THEOREM ON BASE LOCI
II
: 246
10.4 GLOBAL GENERATION OF ADJOINT LINEAR SERIES
II
: 251
10.4.A FUJITA CONJECTURE AND ANGEHRN*SIU THEOREM
II
: 252
10.4.B LOCI OF LOG-CANONICAL SINGULARITIES
II
: 254
10.4.C PROOF OF THE THEOREM OF ANGEHRN AND SIU
II
: 258
10.5 THE EECTIVE NULLSTELLENSATZ
II
: 262
NOTES
II
: 267
11 ASYMPTOTIC CONSTRUCTIONS
II
: 269
11.1 CONSTRUCTION OF ASYMPTOTIC MULTIPLIER IDEALS
II
: 270
11.1.A COMPLETE LINEAR SERIES
II
: 270
11.1.B GRADED SYSTEMS OF IDEALS AND LINEAR SERIES
II
: 276
11.2 PROPERTIES OF ASYMPTOTIC MULTIPLIER IDEALS
II
: 282
11.2.A LOCAL STATEMENTS
II
: 282
11.2.B GLOBAL RESULTS
II
: 285
11.2.C MULTIPLICATIVITY OF PLURIGENERA
II
: 292
11.3 GROWTH OF GRADED FAMILIES AND SYMBOLIC POWERS
II
: 293
XVIII CONTENTS OF VOLUME II
11.4 FUJITA S APPROXIMATION THEOREM
II
: 299
11.4.A STATEMENT AND FIRST CONSEQUENCES
II
: 299
11.4.B PROOF OF FUJITA S THEOREM
II
: 305
11.4.C THE DUAL OF THE PSEUDOEECTIVE CONE
II
: 307
11.5 SIU S THEOREM ON PLURIGENERA
II
: 312
NOTES
II
: 320
REFERENCES
II
: 323
GLOSSARY OF NOTATION
II
: 357
INDEX
II
: 363
|
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| discipline | Mathematik |
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| series2 | Ergebnisse der Mathematik und ihrer Grenzgebiete : 3. Folge |
| spelling | Lazarsfeld, Robert 1953- Verfasser (DE-588)102154279 aut Positivity in algebraic geometry Robert Lazarsfeld Berlin ; Heidelberg ; New York Springer [2004] txt rdacontent n rdamedia nc rdacarrier Ergebnisse der Mathematik und ihrer Grenzgebiete : 3. Folge Algebraische Geometrie (DE-588)4001161-6 gnd rswk-swf Positive Definitheit (DE-588)4382343-9 gnd rswk-swf Algebraische Geometrie (DE-588)4001161-6 s Positive Definitheit (DE-588)4382343-9 s DE-604 DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012869758&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lazarsfeld, Robert 1953- Positivity in algebraic geometry Algebraische Geometrie (DE-588)4001161-6 gnd Positive Definitheit (DE-588)4382343-9 gnd |
| subject_GND | (DE-588)4001161-6 (DE-588)4382343-9 |
| title | Positivity in algebraic geometry |
| title_auth | Positivity in algebraic geometry |
| title_exact_search | Positivity in algebraic geometry |
| title_full | Positivity in algebraic geometry Robert Lazarsfeld |
| title_fullStr | Positivity in algebraic geometry Robert Lazarsfeld |
| title_full_unstemmed | Positivity in algebraic geometry Robert Lazarsfeld |
| title_short | Positivity in algebraic geometry |
| title_sort | positivity in algebraic geometry |
| topic | Algebraische Geometrie (DE-588)4001161-6 gnd Positive Definitheit (DE-588)4382343-9 gnd |
| topic_facet | Algebraische Geometrie Positive Definitheit |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012869758&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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