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Lojasiewicz S. — Introduction to Complex Analytic Geometry
Lojasiewicz S. — Introduction to Complex Analytic Geometry

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Название: Introduction to Complex Analytic Geometry

Автор: Lojasiewicz S.


The subject of this book is analytic geometry, understood as the geometry of analytic sets (or, more generally, analytic spaces), i.e. sets described locally by systems of analytic equations. Though many of the results presented are relatively modern, they are already part of the classical tool-kit of workers in analytic and algebraic geometry and in analysis, for example: the theorems of Chevalley on constructible sets, of Remmert-Stein on removable singularities, of Andreotti-Stoll on the fibres of a finite mapping, and of Andreotti-Salmon on factoriality of the Grassmannian. The chapter on the relationship between analytic and algebraic geometry is particularly illuminating. This book should be regarded as an introduction. Its aim is to familiarize the reader with a basic range of problems, using means as elementary as possible. At the same time, the author's intention is to give the reader accesss to complete proofs without the need to rely on so-called 'well-known' facts. All the necessary properties and theorems have been gathered in the first chapters – either with proofs or with references to standard and elementary textbooks.

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 1991

Количество страниц: 523

Добавлена в каталог: 09.06.2006

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Предметный указатель
$F_{\sigma}$-set      73
$G'_k$ spaces (an algebraic manifold structure)      483
$G'_k$ spaces (an analytic manifold structure)      370
$G'_k$ spaces (topology)      93 94
$G_k(P)$ spaces (an algebraic manifold structure)      483
$G_k(P)$ spaces (an analytic manifold structure)      365
$G_k(P)$ spaces (topology)      96 97
($\Lambda$-) distinguished subspace with respect to a lattice $\Lambda$)      130
($\Lambda$-)homomorphism      1
($\Lambda$-)homomorphism of free modules      10
($\Lambda$-)isomorphism      1
(*-)covering      301
(c-)holomorphic mapping of analytic spaces      311
(k-)complele sequence of coordinate systems      211
(k-)normal (ideal or garm) in a coordinate system      186
(k-)normal analytic germ      186
(k-)normal ideal (in the ring $O_n$)      181
(k-)regular (ideal or germ) in a coordinate system      186
(k-)regular analytic germ      186
(k-)regular ideal (in the ring $O_n$)      186
(k-)regular ideal (in the ring $P_n$)      400
(l-)bounded mapping, near a set      332
(l-)bounded subset of a manifold      183
(l-)bounded subset of an analytic space      332
(N-)homogeneous of degree r holomorphic function      127
(p-) *-covering      301
(p-)sheeted covering      78
(X-)algebraic subset      387
(X-)conic set      386
(X-)subset of a multiprojective space      383
(X-)subset of a projective space      382
(X-)subset of a projective space, defined by homogeneous polynomials      382
(X-)subset of an affine space      164
(X-)subset of an algebraic space      476
(Z-)open subset of a quasi-algebraic set      473
(Z-)open subset of an algebraic space      478
Abelian manifolds      368
Abhyankar lemma      330
Adherence of a quasi-cover      180
Algebra of sets, germs      261
Algebraic alias      474
Algebraic alias of an algebraic space      474
Algebraic element      16
Algebraic extension      20
Algebraic manifold      478
Algebraic set in a quasi-algebraic subset      473
Algebraic space (complex)      474
Algebraic space, a compact analytic space is an      485
Algebraically dependent functions (in $O'_M$)      446
Almost uniform convergence      104
Analytic function      101
Analytic germ      154 282
Analytic germ in a coordinate system      155
Analytic graph theorem      256
Analytic mapping      102
Analytic space      275
Analytic space, reduced      275
Analytic subset of a manifold      155
Analytic subset of an analytic space      277
Analytic subspace      277
Analytically constructive leaf      246
Analytically constructive set      248
Analytically dependent functions (in $O'_M$)      445
Andreotti — Salmon theorem      406 601
Andreotti — Stoll theorem      302
Artin — Rees lemma      36
Artinian module and ring      53
Associated elements of a ring      27
Associated ideals, with a module      38
Associated ideals, with an ideal      34
Atlas complex (modelled on $C^n$)      113
Atlas of an analytic space, analytic      275
Atlas of an analytic space, in the restricted sense, analytic      275
Atlas, analytic      276
Atlas, generalised      114
Atlas, inverse      114
Atlas, maximal      113
Atlas, of a manifold      113
Basic symmetric polynomials      23
Basis of a lattice      130
Bezout theorem      432 435
Biholomorphic (isomorphic) analytic germs      231
Biholomorphic (isomorphic) analytic spaces      281
Biholomorphic at a point, a mapping is      119 273
Biholomorphic manifolds      118
Biholomorphic mapping      107
Biholomorphic mapping manifolds      118
Biholomorphic mapping of analytic spaces      281
Biholomorphic mapping of locally analytic sets      231 272
Biholomorphic mappings of factorial sets, theorem about      480
Biholomorphic mappings of projective spaces, theorem about      437
Biholomorphic sets      272
Blowing-up at a point      373
Blowing-up by means of a family of ideals      376
Blowing-up by means of functions, (elementary)      372
Blowing-up in an (analytic) subset      378
Blowing-up, trivial      372 370
Blown-up apace      372 376 378
Border of a set      73
Branched covering      301
C-linear      08
Canonical basis      9
Canonical charts of the apace $P_n$      367
Canonical coordinate system in $C^n$      116
Cartan — Remmert — Stein lemma      166
Cartan's algebraic characterization of $O_a$      336
Cartan's closed nets theorem      310
Cartan's Lemma      316 467
Cartan's theorem about coherence      214 324 326
Cartesian coordinate systems in $P_n$      357
Cartesian product of analytic spaces      276
Cartesian product of germs (of functions)      82
Cartesian product of manifolds      116
Cartesian product of mappings      14
Cartesian product of rings      2
Catran — Remmert theorem      271
Cauchy — Riemann equations      00
Cauchy's estimates      101
Centre of a blowing-up      372 376
Characteristic polynomial      307
Characteristic zero, ring of      7
Chart at a point      113
Chart of a manifold      113
Chart of an algebraic apace      474
Chart of an analytic apace      276
Chart, genaraliaed, with a linear (affine) model      114
Chart, inverse      114
Chevalley — Reimann theorem      201 311
Chevalley's Theorem      306
Chow's Lemma      483
Chow's theorem in a multiprojective space      384
Chow's theorem in a projective space      382
Chow's theorem in algebraic spaces      485
Chow's theorem on biholomorphic mappings of Grassmann spaces      604
Chow's theorem with a 'parameter'      385
Closed mapping      76
Codimension of a germ of a set      152
Codimension of a submanifold      116
Codimension of a subset of a submanifold      150
Codimension of a vector space      10
Cohen — Macauley ring      66
Coherent family of homeomorphisms (from $O^p_s$ to $O^q_s$)      318
Coherent family of ideals (of the rings $R_s$)      480
Coherent family of subnioditles (of modules $O^m_s$)      324
Collector      170
Compatibility of a family of sets with a family of sets      246
Complete elements of the set $O'_M$ of nearly everywhere holomorphic functions      438
Complete intersection      320
Complete intersection, set-theoretic      202
Composition of germs (of a function and a mapping)      82
Condition (r)      362
Cone      14
Cone of a subset of a projective space      03
Constant dimension, analytic germ of      164
Constant dimension, set of      151 278
Constructible leaf      308
Constructible stratification      300
Constructible subset of a (multi)projective space      302
Constructible subset of a vector space      303
Constructible subset of an algebraic space      476
Coordinate system, (of a manifold)      113
Coordinate system, affine      20
Coordinate system, at a point      113
Coordinate system, linear      20
Cotangent space      41
Covering      77
Crammer's identities      17
Crammer's theorem      10
Crown of a normal triple      203
Decomposition into forms      16 22
Decomposition into simple germs      163
Decomposition of a (locally) analytic set into simple components      217 270
Decomposition of a (locally) analytic set into simple components of constant dimension      218 210
Decomposition of an analytic space into simple components      270
Defined by functions, a (globally analytic) set      163
Defining ideal (in a local noetherian ring)      43
Degree of a polynomial      15
Degree of a polynomial mapping      20
Degree of a polynomial on a vector space      20
Degree of a polynomial with respect to a group of indeterminates      15
Degree of a principal set      412—415
Degree of an algebraic set of constant dimension      416—420
Degree of an element of $Q_n$      142
Denominator of an element (in a ring)      9
Denominator, universal (of an extension of a ring)      9
Depth of a module      63
Derivative of a germ of a holomorphic runction      140
Derivative of a polynomial      10
Diagonal product of germs (of functions)      82
Diagonal product of mappings      14
Differential of a germ of a holomorphic function      140
Differential, (complex)      00 102 121
Dimension (degree) of a field over a field      26
Dimension of a germ of a set      162
Dimension of a set at a point      161 278
Dimension of a submanifold      116
Dimension of a subset of a manifold      160
Dimension of a subset of an analytic space      278
Direction vector of a vector space      13
Discrete valuation ring      44
Discriminant      100
Discriminant of a monic element of $Q_n$      142
Discriminant of a monic polynomial      26
Distinguished germ of degree k      146
Distinguished polynomial      100
Elliptic functions      442—443
Embedding      122
Embedding dimension      41
Embedding of algebraic spaces, (regular)      478
Empty germ (of a set)      80
Equivalent algebraic atlases      474
Equivalent analytic atlases      276
Equivalent analytic atlases (in the restricted sense)      276
Equivalent complex atlases      113
Exceptional set of a *-covering      301
Exceptional set of a blowing-up      372 377
Exceptional set of a nearly everywhere holomorphic function      438
Extended plane      128 120
Extensible holomorphically across a point, a holomorphic function      160
Extension of a ring (field)      1
Extension of an ideal (in a localization)      46 49
Factorial algebraic set      420 487
Factorial ring (unique factorization domain)      27
Factoriality of a noetherian integral domain, characterization of      37
Fiat extension of a noetherian ring      38
Fiat extension of a ring      10
Field of fractions in a field      8
Field of meromorphic germs      440
Field of rational functions on an algebraic set      445
Finite covering      78
Finite covering (characterization of)      78
Finite extension of a field      26
Finite holomorphic mapping      300
Finite ring over a ring      31
Finitely generated (finite) module over a ring      3
Finitely generated ideal      4
Form (of degree k)      15
Form (of degree k) on a vector apace      22
Free module      9
Full germ of a set      80
Function in a coordinate system      20
Gauss' lemma      28
Gauss' theorem      28
Generated cone (by a set)      14
Generated subfleld      2
Generated subfleld, by elements, over a subfield      20
Generated submodule      3
Generated subring      2
Generated subring, by elements, over a subring      19
Generating (a module)      3
Generators of a family of ideals of rings $R_s$      480
Generators of a family of submodules $O^m_s$, finite system of      317
Generators, system of (for a module)      3
Generic dimension of fibers of a (holomorphic) mapping      270
Germ of a function at a point      83
Germ of a function in a coordinate system      140
Germ of a function on a germ of a set      81
Germ of a homeomorphism of germs of sets      82
Germ of a mapping of germs of sets      81
Germ of a set      80
Germ, biholomorphic      231
Globally analytic subset      153
Gluing of (finite) algebraic spaces by isomorphisms      478 479
Gluing of a family of analytic spaces (by biholomorphic mappings)      281
Gluing of a family of manifolds (by biholomorphic mappings)      120 121
Graded ring      66
Grassmann cone      360
Grassmann space (topology)      87—93
Grassmann space as a complex manifold      364
Grassmann space as an algebraic manifold      482
Grassmannian, embedded      368
Grauert — Reimmert formula      307
Grauert — Reimmert lemma      338
Grauert — Reimmert theorem      320
Hadamard's lemma      106
Hadamard's remainder in Taylor's formula      106
Hartogs' lemma      107
Hartogs' theorem      108
Hausdorff space      73
Height of an ideal      56
Hensel's lemma      158
Hilbert — Serre theorem (characterization of regularity of a ring by its homological dimension)      70
Hilbert's basis theorem      30
Hilbert's Nullstellensatz for $R_a$      460
Hilbert's Nullstellensatz for holomorphic germs      165 106 284
Hilbert's Nullstellensatz for polynomials      404
Hilbert's Nullstellensatz in R(V)      429
Hilbert's Nullstellensatz, 'mixed' version      407
Hironaka's construction of spaces which cannot be embedded in a projective space      421—425
Hironaka's example of a non-projective algebraic manifold      487
Hodge's identity      493
Hodge's lemma      406
Hodge's ring      402
Hodge's theorem      493
Holomorphic function      100 101
Holomorphic function at a point      100
Holomorphic function on a locally analytic set      225
Holomorphic function on a manifold      117
Holomorphic function on an analytic space      280
Holomorphic germ of a mapping of an (analytic germ)      230
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