Geometric invariant theory and decorated principal bundles:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Zürich
European Mathematical Society
2008
|
| Schriftenreihe: | Zurich lectures in advanced mathematics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | VII, 389 S. graph. Darst. |
| ISBN: | 9783037190654 |
Internformat
MARC
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| 245 | 1 | 0 | |a Geometric invariant theory and decorated principal bundles |c Alexander H. W. Schmitt |
| 264 | 1 | |a Zürich |b European Mathematical Society |c 2008 | |
| 300 | |a VII, 389 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Zurich lectures in advanced mathematics | |
| 650 | 0 | 7 | |a Geometrische Invariantentheorie |0 (DE-588)4156712-2 |2 gnd |9 rswk-swf |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-016698406 | ||
Datensatz im Suchindex
| _version_ | 1804137959311540224 |
|---|---|
| adam_text | Contents
Introduction
.................................... 1
1 Geometrie Invariant
Theory
......................... 17
1.1
Algebraic Groups and their Representations
.............. 17
1.1.1
Linear Algebraic Groups I
—
Definitions
........... 17
1.1.2
Representations
......................... 19
1.1.3
Linear Algebraic Groups II
—
Linear Algebraic Groups as
Subgroups of General Linear Groups
............. 21
1.1.4
Reductive
Affine
Algebraic Groups
.............. 23
1.1.5
Homogeneous Representations of the General Linear Group
. 24
1.1.6
Faithful Representations and Extensions of Representations
. 26
1.1.7
Appendix. The Reductivity of the Classical Groups via
Weyl s Unitarian Trick
..................... 27
1.2
Geometric Invariant Theory for
Affine
Varieties
—
A First Encounter
30
1.2.1
The Categorical Quotient of a Vector Space
by a Representation
....................... 30
1.3
Examples from Classical Invariant Theory
............... 34
1.3.1
Algebraic Forms
........................ 34
1.3.2
Examples
............................ 35
1.3.3
The Invariant Theory of Matrices
............... 43
1.4
Mumforď s
Geometric Invariant Theory
................ 49
1.4.1
Good and Geometric Quotients
................ 49
1.4.2
Quotients of
Affine
Varieties
.................. 51
1.4.3
Linearizations
.......................... 55
1.5
Criteria for Stability and Semistability
................. 66
1.5.1
The Hilbert-Mumford Criterion
................ 66
1.5.2
Semistability for Direct Sums of Representations
....... 86
1.5.3
Semistability for Actions of Direct Products of Groups
.... 89
1.6
The Variation of GIT-Quotients
..................... 92
1.6.1
The Finiteness of the Number of GIT-Quotients
........ 92
1.6.2
Variation of the GIT-Quotients
................. 93
1.7
The Analysis of Unstable Points
.................... 95
1.7.1
A Few Words about
GIT on Non-
Algebraically Closed Fields
96
1.7.2
The Theory of the Instability Flag
............... 97
1.7.3
The Instability Flag in a Product
................ 102
VI
2
Decorated
Principal
Bundles
......................... 103
2.1.1
Principal
Bundles
—
Definitions and
First
Properties
..... 103
2.1.2
The Classification Problem for Decorated Principal Bundles
. 115
2.2
Rudiments of the Theory of Vector Bundles
.............. 118
2.2.1
The Topology of Vector Bundles
................ 118
2.2.2
The Riemann-Roch Theorem
................. 119
2.2.3
Bounded Families of Vector Bundles
............. 121
2.2.4
The Moduli Space of Semistable Vector Bundles
....... 124
2.3
Decorated Vector Bundles:
Projective
Fibers
.............. 135
2.3.1
Set-Up of the Moduli Problem
................. 135
2.3.2
Semistability of Swamps
.................... 138
2.3.3
Examples
............................ 143
2.3.4
Boundedness
.......................... 144
2.3.5
The Parameter Space
...................... 146
2.3.6
On the Geometry of the Moduli Spaces
............ 163
2.3.7
The Chain of Moduli Spaces
.................. 167
2.4
Principal Bundles as Swamps
...................... 174
2.4.1
Principal Bundles and Associated Vector Bundles
....... 174
2.4.2
Back to Some GIT
....................... 179
2.4.3
Pseudo
G-Bundles
....................... 185
2.4.4
Semistable Reduction for Principal Bundles and the Proof of
Theorem
2.4.1.8 ........................ 188
2.4.5
The Proof of Theorem
2.4.3.3................. 190
2.4.6
The Geometry of the Moduli Spaces
—
A Guide to the Literature
.................... 195
2.4.7
Appendix I: Some Remarks Concerning the Moduli Stack of
Principal Bundles
........................ 195
2.4.8
Appendix II: Moduli Spaces for Principal Bundles
with Reductive Structure Group
via the
Ramanathan-Gómez-Sols
Method
........... 199
2.4.9
Appendix III: Anti-Dominant Characters
........... 208
2.5
Decorated Tuples of Vector Bundles:
Projective
Fibers
............................ 210
2.5.1
Homogeneous Representations
................. 210
2.5.2
More on One Parameter Subgroups
.............. 211
2.5.3
The Moduli Problem of Tumps
................. 217
2.5.4
Proof of Theorem
2.5.3.7.................... 223
2.5.5
Properties of the Semistability Concept
............ 229
2.5.6
Quiver Representations
..................... 237
2.6
Principal Bundles as Tumps
...................... 257
2.6.1
Principal Bundles and Associated Tuples of Vector Bundles
. 258
2.6.2
The Relevant GIT-Quotients
.................. 259
2.6.3
Pseudo
G-Bundles
....................... 263
2.7
Decorated Principal Bundles:
Projective
Fibers
............ 266
2.7.1
The Moduli Spaces
....................... 266
2.7.2
Decorated
Pseudo
G-Bundles
................. 270
vu
2
Decorated
Principal
Bundles
......................... 103
2.7.3
Asymptotic Semistability
.................... 279
2.7.4
Hitchin Pairs
.......................... 283
2.7.5
Fine Tuning of the Theory
................... 286
2.8
Decorated Principal Bundles:
Affine
Fibers
.............. 288
2.8.1
The Moduli Functors
...................... 288
2.8.2
Comparison with
Projective
ρ
-Bumps
.............
292
2.8.3
Construction of the Moduli Spaces
............... 295
2.8.4
Further Properties and Examples
................ 303
2.9
More Generalizations
.......................... 321
2.9.1
Positive Characteristic
..................... 321
2.9.2
Higher Dimensional Base Varieties
.............. 325
2.9.3
Parabolic Structures
...................... 354
Bibliography
................................. 363
Index
..................................... 379
|
| adam_txt |
Contents
Introduction
. 1
1 Geometrie Invariant
Theory
. 17
1.1
Algebraic Groups and their Representations
. 17
1.1.1
Linear Algebraic Groups I
—
Definitions
. 17
1.1.2
Representations
. 19
1.1.3
Linear Algebraic Groups II
—
Linear Algebraic Groups as
Subgroups of General Linear Groups
. 21
1.1.4
Reductive
Affine
Algebraic Groups
. 23
1.1.5
Homogeneous Representations of the General Linear Group
. 24
1.1.6
Faithful Representations and Extensions of Representations
. 26
1.1.7
Appendix. The Reductivity of the Classical Groups via
Weyl's Unitarian Trick
. 27
1.2
Geometric Invariant Theory for
Affine
Varieties
—
A First Encounter
30
1.2.1
The Categorical Quotient of a Vector Space
by a Representation
. 30
1.3
Examples from Classical Invariant Theory
. 34
1.3.1
Algebraic Forms
. 34
1.3.2
Examples
. 35
1.3.3
The Invariant Theory of Matrices
. 43
1.4
Mumforď s
Geometric Invariant Theory
. 49
1.4.1
Good and Geometric Quotients
. 49
1.4.2
Quotients of
Affine
Varieties
. 51
1.4.3
Linearizations
. 55
1.5
Criteria for Stability and Semistability
. 66
1.5.1
The Hilbert-Mumford Criterion
. 66
1.5.2
Semistability for Direct Sums of Representations
. 86
1.5.3
Semistability for Actions of Direct Products of Groups
. 89
1.6
The Variation of GIT-Quotients
. 92
1.6.1
The Finiteness of the Number of GIT-Quotients
. 92
1.6.2
Variation of the GIT-Quotients
. 93
1.7
The Analysis of Unstable Points
. 95
1.7.1
A Few Words about
GIT on Non-
Algebraically Closed Fields
96
1.7.2
The Theory of the Instability Flag
. 97
1.7.3
The Instability Flag in a Product
. 102
VI
2
Decorated
Principal
Bundles
. 103
2.1.1
Principal
Bundles
—
Definitions and
First
Properties
. 103
2.1.2
The Classification Problem for Decorated Principal Bundles
. 115
2.2
Rudiments of the Theory of Vector Bundles
. 118
2.2.1
The Topology of Vector Bundles
. 118
2.2.2
The Riemann-Roch Theorem
. 119
2.2.3
Bounded Families of Vector Bundles
. 121
2.2.4
The Moduli Space of Semistable Vector Bundles
. 124
2.3
Decorated Vector Bundles:
Projective
Fibers
. 135
2.3.1
Set-Up of the Moduli Problem
. 135
2.3.2
Semistability of Swamps
. 138
2.3.3
Examples
. 143
2.3.4
Boundedness
. 144
2.3.5
The Parameter Space
. 146
2.3.6
On the Geometry of the Moduli Spaces
. 163
2.3.7
The Chain of Moduli Spaces
. 167
2.4
Principal Bundles as Swamps
. 174
2.4.1
Principal Bundles and Associated Vector Bundles
. 174
2.4.2
Back to Some GIT
. 179
2.4.3
Pseudo
G-Bundles
. 185
2.4.4
Semistable Reduction for Principal Bundles and the Proof of
Theorem
2.4.1.8 . 188
2.4.5
The Proof of Theorem
2.4.3.3. 190
2.4.6
The Geometry of the Moduli Spaces
—
A Guide to the Literature
. 195
2.4.7
Appendix I: Some Remarks Concerning the Moduli Stack of
Principal Bundles
. 195
2.4.8
Appendix II: Moduli Spaces for Principal Bundles
with Reductive Structure Group
via the
Ramanathan-Gómez-Sols
Method
. 199
2.4.9
Appendix III: Anti-Dominant Characters
. 208
2.5
Decorated Tuples of Vector Bundles:
Projective
Fibers
. 210
2.5.1
Homogeneous Representations
. 210
2.5.2
More on One Parameter Subgroups
. 211
2.5.3
The Moduli Problem of Tumps
. 217
2.5.4
Proof of Theorem
2.5.3.7. 223
2.5.5
Properties of the Semistability Concept
. 229
2.5.6
Quiver Representations
. 237
2.6
Principal Bundles as Tumps
. 257
2.6.1
Principal Bundles and Associated Tuples of Vector Bundles
. 258
2.6.2
The Relevant GIT-Quotients
. 259
2.6.3
Pseudo
G-Bundles
. 263
2.7
Decorated Principal Bundles:
Projective
Fibers
. 266
2.7.1
The Moduli Spaces
. 266
2.7.2
Decorated
Pseudo
G-Bundles
. 270
vu
2
Decorated
Principal
Bundles
. 103
2.7.3
Asymptotic Semistability
. 279
2.7.4
Hitchin Pairs
. 283
2.7.5
Fine Tuning of the Theory
. 286
2.8
Decorated Principal Bundles:
Affine
Fibers
. 288
2.8.1
The Moduli Functors
. 288
2.8.2
Comparison with
Projective
ρ
-Bumps
.
292
2.8.3
Construction of the Moduli Spaces
. 295
2.8.4
Further Properties and Examples
. 303
2.9
More Generalizations
. 321
2.9.1
Positive Characteristic
. 321
2.9.2
Higher Dimensional Base Varieties
. 325
2.9.3
Parabolic Structures
. 354
Bibliography
. 363
Index
. 379 |
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| id | DE-604.BV035029404 |
| illustrated | Illustrated |
| index_date | 2024-07-02T21:49:17Z |
| indexdate | 2024-07-09T21:20:35Z |
| institution | BVB |
| isbn | 9783037190654 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016698406 |
| oclc_num | 254827754 |
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| physical | VII, 389 S. graph. Darst. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | European Mathematical Society |
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| series2 | Zurich lectures in advanced mathematics |
| spelling | Schmitt, Alexander H. W. Verfasser (DE-588)1018798447 aut Geometric invariant theory and decorated principal bundles Alexander H. W. Schmitt Zürich European Mathematical Society 2008 VII, 389 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Zurich lectures in advanced mathematics Geometrische Invariantentheorie (DE-588)4156712-2 gnd rswk-swf Bündel Mathematik (DE-588)4193459-3 gnd rswk-swf Geometrische Invariantentheorie (DE-588)4156712-2 s Bündel Mathematik (DE-588)4193459-3 s DE-604 Erscheint auch als Schmitt, Alexander H. W. Geometric invariant theory and decorated principal bundles Online-Ausgabe 978-3-03719-565-9 (DE-604)BV036758527 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016698406&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Schmitt, Alexander H. W. Geometric invariant theory and decorated principal bundles Geometrische Invariantentheorie (DE-588)4156712-2 gnd Bündel Mathematik (DE-588)4193459-3 gnd |
| subject_GND | (DE-588)4156712-2 (DE-588)4193459-3 |
| title | Geometric invariant theory and decorated principal bundles |
| title_auth | Geometric invariant theory and decorated principal bundles |
| title_exact_search | Geometric invariant theory and decorated principal bundles |
| title_exact_search_txtP | Geometric invariant theory and decorated principal bundles |
| title_full | Geometric invariant theory and decorated principal bundles Alexander H. W. Schmitt |
| title_fullStr | Geometric invariant theory and decorated principal bundles Alexander H. W. Schmitt |
| title_full_unstemmed | Geometric invariant theory and decorated principal bundles Alexander H. W. Schmitt |
| title_short | Geometric invariant theory and decorated principal bundles |
| title_sort | geometric invariant theory and decorated principal bundles |
| topic | Geometrische Invariantentheorie (DE-588)4156712-2 gnd Bündel Mathematik (DE-588)4193459-3 gnd |
| topic_facet | Geometrische Invariantentheorie Bündel Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016698406&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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