Mixed Hodge structures and singularities:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
1998
|
| Edition: | 1. publ. |
| Series: | Cambridge tracts in mathematics
132 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXI, 186 S. |
| ISBN: | 0521620600 |
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MARC
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| 100 | 1 | |a Kulikov, Valentin S. |d 1948- |e Verfasser |0 (DE-588)173143164 |4 aut | |
| 245 | 1 | 0 | |a Mixed Hodge structures and singularities |c Valentine S. Kulikov |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 1998 | |
| 300 | |a XXI, 186 S. | ||
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Record in the Search Index
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| adam_text | Contents
Introduction page xi
I The Gauss Manin connection 1
1 Milnor fibration, Picard Lefschetz monodromy transforma¬
tion, topological Gauss Manin connection 1
1.1 Milnor fibration 1
1.2 Cohomological Milnor fibration 1
1.3 Topological Gauss Manin connection 2
1.4 Picard Lefschetz monodromy transformation 2
2 Connections, locally constant sheaves and systems of linear
differential equations 3
2.1 Connection as a covariant differentiation 3
2.2 Equivalent definition: a covariant derivative along a
vector field 4
2.3 Local calculation of connections. Relation to differen¬
tial equations 5
2.4 The integrable connections. The De Rham complex 6
2.5 Local systems and integrable connections 7
2.6 Dual local systems and connections 8
3 De Rham cohomology 10
3.1 The Poincare lemma 10
3.2 Relative De Rham cohomology 11
3.3 De Rham cohomology for smooth Stein morphisms 11
3.4 Coherence theorem 12
3.5 On the absence of torsion in the De Rham cohomology
sheaves 12
3.6 Relation between .^(/^Q/) and f*.#P(Qy) 13
4 Gauss Manin connection on relative De Rham cohomology 14
vi Contents
4.1 Identification of sheaves of sections of cohomological
fibration and of relative De Rham cohomology 15
4.2 Calculation of the connection on a relative De Rham
cohomology sheaf 16
4.3 The division lemma. The connections on the sheaves
.M^K(X/S) for p =s n 1 17
4.4 The sheaf JV = /*Q£/s/d(/*Q£^) 19
4.5 Meromorphic connections 20
4.6 The Gauss Manin connection as a connecting
homomorphism 21
5 Brieskorn lattices 23
5.1 Brieskorn lattice .M 24
5.2 Calculation of the Gauss Manin connection V on 3@ 25
5.3 Increasing filtration on JZ {Q) 25
5.4 A practical method of calculation of the Gauss Manin
connection 27
5.5 Calculation of the Gauss Manin connection of quasi
homogeneous isolated singularities 28
6 Absence of torsion in sheaves ,X(~l) of isolated
singularities 30
6.1 The presence of a connection implies the absence of
torsion 30
6.2 A theorem of Malgrange 31
6.3 Connection on a pair (E, F) 32
6.4 Sheaves .M(~p) are locally free 32
7 Singular points of systems of linear differential equations 33
7.1 Differential equations of Fuchsian type 33
7.2 Systems of linear differential equations and
connections 34
7.3 Decomposition of a fundamental matrix Y(t) 35
7.4 Regular singular points 36
7.5 Simple singular points 36
7.6 Simple singular points are regular 37
7.7 Connections with regular singularities 39
7.8 Residue and limit monodromy 41
8 Regularity of the Gauss Manin connection 42
8.1 The period matrix and the Picard Fuchs equation 42
8.2 The regularity theorem follows from Malgrange s
theorem 44
8.3 The regularity theorem and connections with
logarithmic poles 44
9 The monodromy theorem 46
9.1 Two parts of the monodromy theorem 46
Contents vii
9.2 Eigenvalues of monodromy 47
9.3 The size of Jordan blocks 49
9.4 Consequences of the monodromy theorem.
Decomposition of integrals into series 49
10 Gauss Manin connection of a non isolated hypersurface
singularity 51
10.1 De Rham cohomology sheaves 51
10.2 Coherence 52
10.3 Relation between .^p(/*Qy) and/*.^(Qy) 53
10.4 A general method of extension of a singular
connection over the whole disk 53
10.5 The sheaves ¦^¦{_ ) and the Gauss Manin connection
dt:.^l2)^.Vp 54
10.6 The sheaves .3kL and the Gauss Manin connection
dtXl^Jth 56
10.7 A generalization of diagram (5.3.4) 57
11 Limit mixed Hodge structure on the vanishing
cohomology of an isolated hypersurface singularity 60
1 Mixed Hodge structures. Definitions. Deligne s theorem 60
1.1 Pure Hodge structure 60
1.2 Polarised HSs 61
1.3 Mixed Hodge structure 61
1.4 Deligne s theorem 62
2 The limit MHS according to Schmid 62
2.1 Variation of HS: geometric case 62
2.2 Variation of HS: definition 63
2.3 Classifying spaces and period mappings 63
2.4 The canonical Milnor fibre 64
2.5 The Schmid limit Hodge filtration Fs 67
2.6 An interpretation of F$ in terms of the canonical
extension of .W. 69
2.7 The weight filtration of a nilpotent operator 70
2.8 Schmid s theorem 73
3 The limit MHS according to Steenbrink 73
3.1 The limit MHS for projective families: the case of
unipotent monodromy 74
3.2 The limit MHS for projective families: the general
case 75
3.3 Brieskorn construction 77
3.4 Limit MHS on a vanishing cohomology 78
3.5 The weight filtration on H (X^). Symmetry of Hodge
numbers 79
viii Contents
4 Hodge theory of a smooth hypersurface according to
Griffiths Deligne 82
4.1 The Gysin exact sequence 82
4.2 Hodge theory for a complement U = X Y. Hodge
filtration and pole order filtration 83
4.3 De Rham complex of the sheaf B[Y]x and the
cohomology of a hypersurface Y 85
4.4 The case of a smooth hypersurface yin a projective
space ^ = P +1 86
4.5 Generalization to the case of a hypersurface with
singularities 87
5 The Gauss Manin system of an isolated singularity 88
5.1 Hodge theory of a smooth hypersurface in the relative
case 89
5.2 The Gauss Manin differential system 90
5.3 Interpretation of the complex DRZ/S (5[r]z) in terms
of the morphism/: X — S 91
5.4 Connection between the differential system,%x and
the Brieskorn lattice .M{Q) 94
6 Decomposition of a meromorphic connection into a direct
sum of the root subspaces of the operator tdt. The K
filtration and the canonical lattice 95
6.1 Block decomposition 95
6.2 Decomposition of a meromorphic connection. J6 into
a direct sum of the root subspaces 96
6.3 The order function a and the V filtration 98
6.4 Identification of the zero fibre of the canonical
extension /. and the canonical fibre of the fibration H 99
6.5 The decomposition of sections co s . / 6 into a sum of
elementary sections 100
6.6 Transfer of automorphisms from the Milnor lattice H
to the meromorphic connection. /6 101
7 The limit Hodge filtration according to Varchenko and to
Scherk Steenbrink 103
7.1 Motivation of Scherk Steenbrink s construction of the
Hodge filtration 103
7.2 The definition of the limit Hodge filtration Fss
according to Scherk Steenbrink 106
7.3 The Scherk Steenbrink theorem 108
7.4 Varchenko s theorem about the operator of
multipl ication by / in Q f 110
7.5 The definition of the limit Hodge filtration F on
H (X^) according to Varchenko 111
Contents ix
7.6 Comparison of the filiations Fss and FVa 111
7.7 Supplement on the connection between the Gauss
Manin differential system .?/ x and its meromorphic
connection. //C 112
8 Spectrum of a hypersurface singularity 115
8.1 The definition of the spectrum of an isolated singularity 115
8.2 The spectral pairs Spp(f) 117
8.3 Properties of the spectrum 118
8.4 The spectra of a quasihomogeneous and a semi
quasihomogeneous singularity 119
8.5 Calculation of the spectrum of an isolated singularity
in terms of a Newton diagram 122
8.6 Calculation of the geometric genus of a hypersurface
singularity in terms of the spectrum 127
8.7 Spectrum of the join of isolated singularities 127
8.8 Spectra of simple, uni and bimodal singularities 129
8.9 Semicontinuity of the spectrum. Stability of spectrum
for ,« const deformations 130
8.10 Spectrum of a non isolated singularity 132
8.11 Relation between the spectrum of a singularity with a
one dimensional critical set and spectra of isolated
singularities of its Iomdin series 134
III The period map of a /y const deformation of an isolated
hypersurface singularity associated with Brieskorn
lattices and MHSs 139
1 Gluing of Milnor fibrations and meromorphic connections
of a // const deformation of a singularity 139
1.1 Milnor fibrations 140
1.2 Cohomological fibration 141
1.3 Canonical extension of the sheaf . and the
meromorphic connection 142
2 Differentiation of geometric sections and their root
components wrt a parameter 144
2.1 Geometric sections and their root components 144
2.2 Formulae for derivatives of geometric sections and
their root components wrt a parameter 146
2.3 Decomposition of the root components of geometric
sections into Taylor series for upper diagonal deforma¬
tions of quasihomogeneous singularities 148
2.4 The sheaves Gr^10 150
x Contents
3 The period map 151
3.1 Identification of meromorphic connections in a
jM const family of singularities 151
3.2 The period map defined by the embedding of
Brieskorn lattices 152
3.3 Example: the period map for £12 singularities 154
3.4 The period map for hyperbolic singularities Tp^r 156
3.5 The period map for simply elliptic singularities 159
3.6 The period map defined by MHS on the vanishing
cohomology 163
4 The infinitesimal Torelli theorem 165
4.1 The V filtration on Jacobian algebra. The necessary
condition for,« const deformation 165
4.2 Calculation of the tangent map of the period map. The
horizontality of the MHS period map 167
4.3 The infinitesimal Torelli theorem 169
4.4 The period map in the case of quasihomogeneous
singularities 171
5 The Picard Fuchs singularity and Hertling s invariants 172
5.1 The Picard Fuchs singularity PFS(f) according to
Varchenko 172
5.2 The Hertling invariant Herx (/) 174
5.3 The Hertling invariants Her2{f) and Hen(f) 177
5.4 Hertling s results 179
References 181
Index 185
|
| any_adam_object | 1 |
| author | Kulikov, Valentin S. 1948- |
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| dewey-ones | 516 - Geometry |
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| dewey-search | 516.3/5 |
| dewey-sort | 3516.3 15 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 1. publ. |
| format | Book |
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| id | DE-604.BV011930751 |
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| language | English |
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| spelling | Kulikov, Valentin S. 1948- Verfasser (DE-588)173143164 aut Mixed Hodge structures and singularities Valentine S. Kulikov 1. publ. Cambridge [u.a.] Cambridge Univ. Press 1998 XXI, 186 S. txt rdacontent n rdamedia nc rdacarrier Cambridge tracts in mathematics 132 Hodge theory Singularities Mathematics Singularität Mathematik (DE-588)4077459-4 gnd rswk-swf Hodge-Theorie (DE-588)4135967-7 gnd rswk-swf Singularität Mathematik (DE-588)4077459-4 s Hodge-Theorie (DE-588)4135967-7 s DE-604 Cambridge tracts in mathematics 132 (DE-604)BV000000001 132 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008064191&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kulikov, Valentin S. 1948- Mixed Hodge structures and singularities Cambridge tracts in mathematics Hodge theory Singularities Mathematics Singularität Mathematik (DE-588)4077459-4 gnd Hodge-Theorie (DE-588)4135967-7 gnd |
| subject_GND | (DE-588)4077459-4 (DE-588)4135967-7 |
| title | Mixed Hodge structures and singularities |
| title_auth | Mixed Hodge structures and singularities |
| title_exact_search | Mixed Hodge structures and singularities |
| title_full | Mixed Hodge structures and singularities Valentine S. Kulikov |
| title_fullStr | Mixed Hodge structures and singularities Valentine S. Kulikov |
| title_full_unstemmed | Mixed Hodge structures and singularities Valentine S. Kulikov |
| title_short | Mixed Hodge structures and singularities |
| title_sort | mixed hodge structures and singularities |
| topic | Hodge theory Singularities Mathematics Singularität Mathematik (DE-588)4077459-4 gnd Hodge-Theorie (DE-588)4135967-7 gnd |
| topic_facet | Hodge theory Singularities Mathematics Singularität Mathematik Hodge-Theorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008064191&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000001 |
| work_keys_str_mv | AT kulikovvalentins mixedhodgestructuresandsingularities |