Schlichenmaier M. — An Introduction to Riemann Surfaces, Algebraic Curves and Moduli Spaces :: Электронная библиотека попечительского совета мехмата МГУ

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Schlichenmaier M. — An Introduction to Riemann Surfaces, Algebraic Curves and Moduli Spaces

Schlichenmaier M. — An Introduction to Riemann Surfaces, Algebraic Curves and Moduli Spaces

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Название: An Introduction to Riemann Surfaces, Algebraic Curves and Moduli Spaces

Автор: Schlichenmaier M.

Аннотация:

This book gives an introduction to modern geometry. Starting from an elementary level the author develops deep geometrical concepts, playing an important role nowadays in contemporary theoretical physics. He presents various techniques and viewpoints, thereby showing the relations between the alternative approaches.

At the end of each chapter suggestions for further reading are given to allow the reader to study the touched topics in greater detail.

This second edition of the book contains two additional more advanced geometric techniques: (1) The modern language and modern view of Algebraic Geometry and (2) Mirror Symmetry.

The book grew out of lecture courses. The presentation style is therefore similar to a lecture. Graduate students of theoretical and mathematical physics will appreciate this book as textbook. Students of mathematics who are looking for a short introduction to the various aspects of modern geometry and their interplay will also find it useful. Researchers will esteem the book as reliable reference.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1989

Количество страниц: 149

Добавлена в каталог: 22.11.2009

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID

Предметный указатель

$c_{1}$       103
-function      35 72 79
-adic expansion      139
-adic integers      140
-adic numbers      138
-adic order      140
-adic power series      139
-adic valuation      137
-expansion      72
$\check{C}$ ech cohomology      24 94 96
$\check{C}$ ech-cycles      94
Abelian variety      64 99
Abelianization      17
Absolute value      134
Adeles      141
Affine variety      94 97
Algebraic bundles      89 93
Algebraic differentials      67
Algebraic integers      142
Algebraic map      59 60
Algebraic variety      58 94 97
Algebraically closed      135
Analytic isomorphism      9 26
Arithmetic structure      82 134
Associated sheaf      119
Atlas      1
Automorphisms of curves      70 77 81
Base variety      67
Belavin — Knizhnik theorem      123
Betti number      17
Biholomorphic map      9
Boundaries      16
Boundary map      15
Boundary of $\mathcal{M}_{g}$       78
Bundle corresponding to a point      104
Canonical bundle      105
Canonical bundle of the moduli space      122
Canonical divisor      31 41
Canonical divisor class      41 76 111 131
Canonical homology basis      21 51 54
Cauchy sequence      134
Cauchy — Riemann differential equations      4
Chain      95
Chains      15 45
Chern character      127
Chern classes      126
Chern form      06
Chern number      98
Chow ring      126
Chow ring of a point      130
Chow's theorem      60
Classification      7
Closed differential form      44
Closed sets of $\mathbb{P}^{1}$       57
Coarse moduli space      69
Coboundary      96
Coboundary map      95
Cocycle      86 96
Coherent sheaf      93
Cohomologous cocycles      87
Cohomology      94
Cohomology ring      131
Cohomology sequence      97
Commutator subgroup      17
Compactification of the moduli space      77
Comparison theorems      97
Complex manifold      4
complex numbers      135
Complex projective space      4
Coordinate change map      3
Coordinate patch      1
Cotangent bundle      39 76 88
Cotangent bundle the moduli space      122
Covering      22 74
Covering transformation      23
Cubic curve      61 73
Curve defined over the rational numbers      79 134
Curves with elliptic tails      81
Cusp      79
Cuspidal cubic      79
Cycle conditions      86
Cycles      16
De — Rham cohomology group      44
Defining cocycle      87
Degree of a divisor      31
Degree of a line bundle      106
Derivation      37
Derived functor      94 97
diff      75
Diff ${}_{0}$       75
Diffeomorphism      8
Differentiable isomorphism      8
Differentiable manifold      3
Differentiable structures on $\mathbb{R}^{4}$       9
Differential      88
Differential form      38 44 88
Differential form of second order      43
Dimension of a manifold      1
Dimension of a variety      59
Direct image sheaves      119
Discriminant function      35
Divisor      30 41 59 94 99
Divisor class      30
Divisor of a section      102
Double point      78
Doubly periodic meromorphic function      34
Dual of a vector bundle      88
Eisenstein series      35
Elliptic curve      61 73
Embedding of tori      61 63
Euler — Poincar $\acute{e}$ characteristic      19
Exact differential form      44
Exponential sequence      92 98
Exterior differentiation      43
Exterior power      43
Exterior product of a vector bundle      89
Family of curves      67
Family of vector spaces      85
Field extension      142
Field of meromorphic functions      26 29
Fine moduli space      70
Fractional linear transformation      74
Free Abelian group      30
Fundamental group      11 19 23 74
Genus      8 18 31 67
Geometric invariant theory      7 7
Global field      140
Global holomorphic differential      42
Glueing function      3
Grothendieck group      125
Grothendieck — Riemann — Roch theorem      128
Harmonic      44
Haussdorff space      1
Higher dimensional torus      48
Higher direct image sheaves      119
Hirzebruch — Riemann — Roch theorem      131
Hodge bundle      80 81 82 122
Hodge class      80
Holomorphic bundles      89 93
Holomorphic differentials      40 42 47 51 67 76 80
Holomorphic form      40
Holomorphic function      25 28
Holomorphic functional determinant      4
Holomorphic functions on $\mathcal{M}_{g}$       77
Holomorphic map      26 60
Homeomorphism      8
Homogeneous coordinates      5
Homogeneous polynomial      56
Homomorphism of families      85
Homotopy      10

Ideles      141
Integration      45
Integration pairing      46
Intersection product      20
Irreducible variety      58
Isomorphism of principally polarized tori      48
Isomorphy of families      68
Isothermal coordinates      7
Isotopy      74
Jacobi map      53
Jacobian      52
Kodaira's embedding theorem      60
Krichever — Novikov algebra      115
Lattice      33 48
Lie algebra $sl(2, \mathbb{C})$       110
Line bundle      81 86 98 99
Linear variety      58
Local field      140
Local-global principle      141
Locally free resolution      125
Locally free sheaf      93
Loop      10
Mapping class group      75
Meromorphic differential      40 46 60
Meromorphic form      40
Meromorphic function      26 60
Meromorphic function on $\mathbb{P}^{1}$       29
Meromorphic function on the torus      33 34
Meromorphic sections of vector bundles      101 107
Meromorphic vector field      115
Metric      7
Modular forms      82
Modular function      72
Module      88
Moduli space      67 122
Multiplicity of a pole      27
Multiplicity of a zero      27
Mumford isomorphism      122 130
Mumford — Deligne — Knudsen compactification      78
Nodal cubic      79
nodes      78
Noetherian ring      58
Non-archimedian valuation      137
Nonsingular variety      60
Normal variety      77
Number field      83 142
ord      40
Order of a meromorphic function      30
Orient able manifold      6
Orientation      15
Ortsuniformisierende      25
Ostrowski's theorem      137
Paracompact      1
Partition of unity      2
Path      10
Period matrix      51
Picard group      31 80 81
Picard group of $\mathcal{M}_{g}$       80
Picard group of the moduli functor      81 122
Picard variety      53
Polarization      48 64
Polyakov form      124
Polyakov integration measure      123
Polyakov partition function      123
Presheaf      119
Prime form      55
Principal divisor      30
Principally polarized abelian variety      75 82
Principally polarized tori      48 63 75
Projective line $\mathbb{P}^{1}$       5 42 57 70
Projective nonsingular curve      60
Projective space      4 56 94
Projective variety      58 60 99
Proper algebraic map      126
Pullback      48 89
Pullback of a family      69
Quadratic differential      76 112 121
Quasiperiodic function      63
Quasiprojective variety      58
Ramified covering      22
Rational differential      60
Rational equivalence      126
Rational function      59 60
Rational integers      142
Rational point      83 83
Real analytic manifold      3
Real numbers      134
Relative differentials      120
Relative tangent sheaf      120
Residue theorem      41
Residuum      40
Restricted Picard group      31 53
Restriction map      90
Riemann bilinear relations      51
Riemann sphere      5 42
Riemann surface      7 61
Riemann zeta function      142
Riemann — Roch theorem      31 107 131
Satake compactification      77
Schottky problem      75
Section of a vector bundle      87
Separated topological space      1
Serre duality      99 107
Sheaf      90
Sheaf cohomology      24 94
Sheaf homomorphism      91
Sheaf of -differential forms      99
Sheaf of $\mathcal{O}$ -modules      91 93
Sheaf of differentiable functions      91
Sheaf of holomorphic functions      91
Sheaf of locally constant functions      90
Sheaf of regular functions      91
Sheaf of sections      90 93
Short exact sequence of sheaves      92 97
Siegel upper-half-space      62 75
simplex      15 45
Simplicial homology group      16
Simply connected      12
Singular homology      24
Singular point      60 61
Singularities of $\mathcal{M}_{g}$       77
Smooth variety      60
Sphere $S^{2}$       12 18
Sphere $S^{7}$       9
Sphere $S^{n}$       3
Stable curves      77
Standard coordinates      25
Stokes' theorem      45
Structure sheaves      91
Subvarieties      94
Sum of vector bundles      88
Symplectic basis      49
Symplectic group      50 54 72 75 82
Tangent bundle      39 76 85 88
Tangent bundle the moduli space      122
Tangent space      37
Tangent space of $\mathcal{M}_{g}$       76
Teichm $\ddot{u}$ ller space      74
Teichm $\ddot{u}$ ller surface      74
Tensor      89
Tensor product of vector bundles      88
Theta function      53 62 99
Theta function with characteristic      64
Theta null values      82
Todd class      128
Topological genus      18
Topological isomorphism      8
Topological normal form      17
Torelli's theorem      53 75
Torus      12 18 33 42 48 71

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