Differential topology of complex surfaces: elliptic surfaces with pg = 1: smooth classification
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
1993
|
| Series: | Lecture notes in mathematics
1545 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | VII, 224 S. |
| ISBN: | 3540566740 0387566740 |
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| 100 | 1 | |a Morgan, John W. |d 1946- |e Verfasser |0 (DE-588)129352446 |4 aut | |
| 245 | 1 | 0 | |a Differential topology of complex surfaces |b elliptic surfaces with pg = 1: smooth classification |c John W. Morgan ; Kieran G. O'Grady |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 1993 | |
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Record in the Search Index
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| adam_text |
Contents
1 Introduction 1
1.1 Statement of the main results 2
1.2 Background 3
1.3 Outline of the paper 6
1.4 Conventions and notation 8
2 Unstable polynomials of algebraic surfaces 12
2.1 Introduction 12
2.1.1 Generalities on the i/ map 14
2.2 A stratification of parameter spaces for vector bundles on S 15
2.2.1 An inductive procedure that defines the type of a bundle near E 16
2.2.2 Definition of the stratification by type near E 18
2.2.3 The pushforward to 5 19
2.3 The stratification of Mc+k(S, H) 19
2.3.1 The case of polarizations near to v'H 20
2.3.2 The morphism from X* " to Xc+lt_|t|(S, tf) 21
2.4 The At construction 21
2.4.1 The construction of At(V) 22
2.4.2 Proof of Proposition 2.4.1 24
2.5 Analysis of the strata of Mc+k(S, H(r)) 25
2.5.1 The strata X^" 25
2.5.2 The strata X1" 26
2.6 Proofs of the theorems 27
2.6.1 Proof of Theorem 2.1.1 27
2.6.2 Relative moduli spaces 28
2.6.3 The relative v map and the invariance of 6 30
2.6.4 Proof of Theorem 2.1.2 32
3 Identification of £3iI.(S, H) with y3(S) 33
3.1 The main results 33
3.2 The family of K3 surfaces with a section 35
3.2.1 The period space and the global Torelli theorem 35
3.2.2 Construction of the family 37
3.3 The family of minimal elliptic surfaces with multiple fibers 40
3.3.1 Construction of the family 40
3.3.2 Relationship between the cohomology of an elliptic fibration and
that of its jacobian surface 42
3.3.3 Analogue of Theorem 3.2.10 for the family of elliptic surfaces
with multiple fibers 46
3.4 The family of blown up elliptic surfaces 49
3.5 Proof of Theorem 3.1.4 50
3.5.1 The generic subset of T 51
3.5.2 The Invariant theory argument 55
VI
4 Certain moduli spaces for bundles on elliptic surfaces with pg = 1 57
4.1 Background material on extensions of rank one sheaves 58
4.2 The parameter spaces for properly semi stable bundles 59
4.3 The moduli spaces MC(S, H) for 1 c 3 64
4.3.1 A description of V as an extension 64
4.3.2 The parameter spaces for vertical extensions 67
4.3.3 Computation of dimensions of cohomology groups 69
4.3.4 The dimension of M£S, H) 70
4.4 Irreducible components of A43(S,H) associated to large divisors . 73
4.5 Four dimensional components of M2(S, H) 79
4.5.1 A more detailed study of M?(S, H) 82
4.6 Multiplicities 86
4.6.1 The versal deformation space of a vector bundle 87
4.6.2 Proof of Theorem 4.6.1 88
4.6.3 Proof of Theorem 4.6.2 93
4.6.4 Proof of Theorem 4.6.3 95
4.7 Definition of 6£(S, H) and 6£(S, H) 95
4.7.1 6«(S,H) _^ 96
4.7.2 The line bundle Mo over P3,t(5, H) in the case L2 S Os 96
4.7.3 6?(S,H) 97
5 Representatives for classes in the image of the i/ map 99
5.1 Representatives for the v map 99
5.1.1 Generalities on Chern classes 99
5.1.2 A divisor representing f([C]) 99
5.1.3 Holomorphic 2 form representatives 101
5.1.4 The divisors A and two forms A on MC{S,H) and PC(S,H) . . 103
5.1.5 Elementary properties of A^ (C, L) and \jr(u ) 105
5.2 Passage from the blow up to the original surface 106
5.2.1 Relation between the // map for 5 and 5 106
5.2.2 Avoiding base points 107
5.3 Enumerative Geometry 107
5.4 e2(S,H) Ill
6 The blow up formula 112
6.1 Outline of the proof of Theorem 6.0.1 for k = 2 113
6.2 First results 116
6.2.1 The basic properties of the divisors A(£i) and A(E2) 116
6.2.2 The definition of Y(X),c(X),d(X) 117
6.2.3 Comparing classes after semistable reduction 117
6.3 An extension of the family At(V) 120
6.3.1 The basic construction 120
6.3.2 The basic formula 124
6.4 Proof of Proposition 6.1.3 125
6.4.1 Enumerating the components Aj 125
6.4.2 Proof of Proposition 6.4.1 in the case when Y(Aj) C M2(S,H) 128
6.4.3 Proof of Proposition 6.4.1 in the case when Y{A ) C P0(S,H) . 129
6.5 The contribution of the Xi 134
VII
6.5.1 Initial cases when the contribution is zero 135
6.5.2 The X{ such that Y(Xt) C M3{S,H) 137
6.5.3 The X, such that Y(Xi) C M2(S,H) 140
6.5.4 The X, such that Y(Xi) C P3{S,H) 151
6.5.5 The Xi such that Y(X ) C PC(S,H) with c 2 156
6.6 The multiplicity of the X{ such that S(Xi) ^ 0 160
6.6.1 The scheme A0 160
6.6.2 The case when Y(X{) C M3(S, H) 162
6.6.3 The case when Y(Xi) C M2{S, H) 163
6.6.4 The case when Y(Xi) C P3{S,H) 163
7 The proof of Theorem 1.1.1 167
7.1 Only the components of Aii{S, H) associated to large divisors con¬
tribute to the first two coefficients of 6*(S, H) 169
7.2 The proof of the first part of Proposition 7.0.10 171
7.2.1 The combinatorics of the set R 171
7.3 A further study of the components Ml?(S,H) 174
7.3.1 Properties of VD in the n"1 order neighborhood of F$ 176
7.3.2 Certain extensions on 5 and their properties 178
7.3.3 Some local computations 180
7.3.4 The proof of Proposition 7.3.2 182
7.4 The computation of c'j(mi,m2) 189
7.4.1 Reduction to a computation on Hilb3(S) 190
7.4.2 An expression for Vwd([C]) 192
7.4.3 An expression for c/1(m1,m2) as an integral over R 193
7.4.4 A more explicit expression for c^(7n1,m2) 196
7.4.5 Formulas for certain sums over T 199
7.4.6 Completion of the proof of Proposition 7.0.11 201
7.5 Proof of Formula (79 ) and of Proposition 7.0.12 202
7.5.1 Proof of Proposition 7.5.1 202
7.5.2 Proof of Proposition 7.0.12 208
8 Appendix: The non simply connected case by John W. Morgan,
Millie Niss and Kieran O'Grady 211
8.1 Proof of Proposition 8.0.20 212
8.2 Proof of Proposition 8.0.21 214
8.3 Computation oie2(S,H) 216
References 219
Index 222 |
| any_adam_object | 1 |
| author | Morgan, John W. 1946- O'Grady, Kieran G. |
| author_GND | (DE-588)129352446 |
| author_facet | Morgan, John W. 1946- O'Grady, Kieran G. |
| author_role | aut aut |
| author_sort | Morgan, John W. 1946- |
| author_variant | j w m jw jwm k g o kg kgo |
| building | Verbundindex |
| bvnumber | BV007745895 |
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| callnumber-label | QA3 |
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| callnumber-subject | QA - Mathematics |
| classification_rvk | SI 850 |
| classification_tum | MAT 322f MAT 576f |
| ctrlnum | (OCoLC)722128300 (DE-599)BVBBV007745895 |
| dewey-full | 516.3/52 510 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry 510 - Mathematics |
| dewey-raw | 516.3/52 510 |
| dewey-search | 516.3/52 510 |
| dewey-sort | 3516.3 252 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| physical | VII, 224 S. |
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| spelling | Morgan, John W. 1946- Verfasser (DE-588)129352446 aut Differential topology of complex surfaces elliptic surfaces with pg = 1: smooth classification John W. Morgan ; Kieran G. O'Grady Berlin [u.a.] Springer 1993 VII, 224 S. txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics 1545 Algebraïsche meetkunde gtt Differentiaalmeetkunde gtt Surfaces elliptiques Topologie différentielle Differential topology Elliptic surfaces Komplexe algebraische Fläche (DE-588)4164889-4 gnd rswk-swf Elliptische Fläche (DE-588)4152027-0 gnd rswk-swf Differentialtopologie (DE-588)4012255-4 gnd rswk-swf Differentialtopologie (DE-588)4012255-4 s Komplexe algebraische Fläche (DE-588)4164889-4 s DE-604 Elliptische Fläche (DE-588)4152027-0 s O'Grady, Kieran G. Verfasser aut Lecture notes in mathematics 1545 (DE-604)BV000676446 1545 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005093362&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Morgan, John W. 1946- O'Grady, Kieran G. Differential topology of complex surfaces elliptic surfaces with pg = 1: smooth classification Lecture notes in mathematics Algebraïsche meetkunde gtt Differentiaalmeetkunde gtt Surfaces elliptiques Topologie différentielle Differential topology Elliptic surfaces Komplexe algebraische Fläche (DE-588)4164889-4 gnd Elliptische Fläche (DE-588)4152027-0 gnd Differentialtopologie (DE-588)4012255-4 gnd |
| subject_GND | (DE-588)4164889-4 (DE-588)4152027-0 (DE-588)4012255-4 |
| title | Differential topology of complex surfaces elliptic surfaces with pg = 1: smooth classification |
| title_auth | Differential topology of complex surfaces elliptic surfaces with pg = 1: smooth classification |
| title_exact_search | Differential topology of complex surfaces elliptic surfaces with pg = 1: smooth classification |
| title_full | Differential topology of complex surfaces elliptic surfaces with pg = 1: smooth classification John W. Morgan ; Kieran G. O'Grady |
| title_fullStr | Differential topology of complex surfaces elliptic surfaces with pg = 1: smooth classification John W. Morgan ; Kieran G. O'Grady |
| title_full_unstemmed | Differential topology of complex surfaces elliptic surfaces with pg = 1: smooth classification John W. Morgan ; Kieran G. O'Grady |
| title_short | Differential topology of complex surfaces |
| title_sort | differential topology of complex surfaces elliptic surfaces with pg 1 smooth classification |
| title_sub | elliptic surfaces with pg = 1: smooth classification |
| topic | Algebraïsche meetkunde gtt Differentiaalmeetkunde gtt Surfaces elliptiques Topologie différentielle Differential topology Elliptic surfaces Komplexe algebraische Fläche (DE-588)4164889-4 gnd Elliptische Fläche (DE-588)4152027-0 gnd Differentialtopologie (DE-588)4012255-4 gnd |
| topic_facet | Algebraïsche meetkunde Differentiaalmeetkunde Surfaces elliptiques Topologie différentielle Differential topology Elliptic surfaces Komplexe algebraische Fläche Elliptische Fläche Differentialtopologie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005093362&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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