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Lojasiewicz S. — Introduction to Complex Analytic Geometry
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Название: Introduction to Complex Analytic GeometryАвтор: Lojasiewicz S. Аннотация: 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: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 1991Количество страниц: 523Добавлена в каталог: 09.06.2006Операции: Положить на полку |
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73 483 370 93 94 483 365 96 97 ( 130 ( 1 ( 10 ( 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 181 (k-)regular (ideal or germ) in a coordinate system 186 (k-)regular analytic germ 186 (k-)regular ideal (in the ring 186 (k-)regular ideal (in the ring 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 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 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 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 367 Canonical coordinate system in 116 Cartan — Remmert — Stein lemma 166 Cartan's algebraic characterization of 336 Cartan's closed nets theorem 310 Cartan's Lemma 316 467 Cartan's theorem about coherence 214 324 326 Cartesian coordinate systems in 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 318 Coherent family of ideals (of the rings 480 Coherent family of subnioditles (of modules 324 Collector 170 Compatibility of a family of sets with a family of sets 246 Complete elements of the set 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 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 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 480 Generators of a family of submodules 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 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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