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Fritzsche K., Grauert H. — From Holomorphic Functions To Complex Manifolds
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Название: From Holomorphic Functions To Complex ManifoldsАвторы: Fritzsche K., Grauert H. Аннотация: This book is an introduction to the theory of complex manifolds. The authors' intent is to familiarize the reader with the most important branches and methods in complex analysis of several variables and to do this as simply as possible. Therefore, the abstract concepts involving sheaves, coherence, and higher-dimensional cohomology have been completely avoided. Only elementary methods such as power series, holomorphic vector bundles, and one-dimensional cocycles are used. Nevertheless, deep results can be proved, for example, the Remmert-Stein theorem for analytic sets, finiteness theorems for spaces of cross sections in holomorphic vector bundles, and the solution of the Levi problem. Each chapter is complemented by a variety of examples and exercises. The only prerequisite needed to read this book is a knowledge of real analysis and some basic facts from algebra, topology, and the theory of one complex variable. The book can be used as a first introduction to several complex variables as well as a reference for the expert.
Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 2002Количество страниц: 392Добавлена в каталог: 20.11.2009Операции: Положить на полку |
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108 109 357 1-complete 267 1-convex 267 Abel's lemma 12 Abelian variety 351 Absolute space 7 Accessible boundary point 101 Acyclic covering 185 Adjoint operator 330 Adjunction formula 215 Affine algebraic 217 Affine algebraic manifold 224 Affine algebraic set 224 Affine part 210 Algebraic derivative 125 Analytic disk 47 102 Analytic polyhedron 85 Analytic set 36 160 Analytic tangent 275 Analytically dependent 320 Antiholomorphic form 303 Antiholomorphic gradient 28 Ascoli, lemma of 23 Atlas 155 Automorphism 226 Ball 5 Banach algebra 105 Bergman kernel 370 Bergman metric 372 Bergman projection 370 Bertini, theorem of 225 Betti number 190 Biholomorphic 33 159 Blowup 237 boundary conditions 357 Boundary distance 54 58 Boundary function 64 Boundary operator 189 Bracket 332 Branch locus 227 Branch point 128 Branched covering 128 227 Branched domain 229 Bubble method 283 Canonical bundle 174 Caratheodory distance 371 Cartan — Thullen, theorem of 77 Cauchy integral 17 Cauchy — Riemann equations 29 Cech cohomology 185 Center of a modification 236 Chain rule 32 Chern class 256 Chow, Theorem of 217 Closed map 226 Closure of 245 Closure of a manifold 244 Coboundary 184 187 Coboundary operator 183 Cochain 183 187 Cocycle 183 187 Codimension 151 Cohomologous 174 Cohomology group 184 187 Commutator 332 Compact exhaustion 163 Compact map see "Completely continuous" Compactification, one-point 153 Compatibility condition 173 Compatible coordinate systems 154 Complete hull 24 completely continuous 285 Completely singular 49 Complex 1-form 261 Complex covariant r-tensor 261 Complex differentiable 14 Complex gradient 15 Complex manifold 155 Complex n-space 1 Complex structure 1 155 Complexification 2 Cone 216 Conical set 216 Conjugation 2 Connected 6 88 Continuity of roots 134 Continuity principle 47 Continuity theorem 43 Contractible space 189 Convergence, compact 12 Convergence, domain of 13 Convergent series of numbers 10 Convex 8 66 74 Convex hull 73 Convex, strictly 67 Coordinate system 154 Cousin-I distribution 253 Cousin-II distribution 254 Covariant tangent vector 261 Critical locus 227 Cross section 176 cuboid 276 Cuboid, distinguished 276 Current 328 Current, harmonic 336 de Rham group 311 de Rham isomophism 312 de Rham sequence 310 Decomposition theorem of Hodge — Kodaira 339 Defining function 64 Degree of a hypersurface 220 Derivation 33 164 Derivative of a holomorphic map 30 Derivative, directional 16 Derivative, partial 15 Derivative, real 26 Derivatives, Wirtinger 28 Diagonal 162 Differential 260 300 Differential form 297 300 DIMENSION 161 Dimension of a submanifold 39 Dimension of an analytic set 142 Dimension of an irreducible analytic set 141 Dimension, local 150 Dirichlet's principle 52 Discrete map 228 Discriminant 127 Discriminant set 128 Distinguished boundary 6 Divisor 198 Divisor of a meromorphic function 198 Divisor of a meromorphic section 201 Dolbeault group 311 Dolbeault sequence 310 Dolbeault's lemma 309 Dolbeault's theorem 312 Domain 6 Domain of Existence 97 Domain of holomorphy 49 99 Domain of holomorphy, weak 49 Dual bundle 178 Duality theorem of Poincare 337 Duality theorem of Serre — Kodaira 338 Effective divisor see "Positive divisor" Effective group operation 172 Elementary symmetric polynomial 126 Elliptic operator 337 Embedded-analytic irreducible component 135 Embedded-analytic set 135 Embedding 214 Embedding dimension 152 Embedding theorem 145 258 Energy form 359 Envelope of holomorphy 87 97 Equivalence of bundles 174 Equivalence of systems of transition functions 174 Euclidean coordinates 322 Euclidean Domain 125 Euler sequence 223 Exact sequence 223 Exceptional set 236 Exhaustion function 58 Exterior algebra 298 Exterior derivative 301 Factorial monoid 125 Faithful see "Effective group operation" Fefferman, theorem of 369 Fiber bundle 173 Fiber bundle isomorphism 174 Fiber metric 341 Fiber product 171 Filter basis 100 Finite map 227 Finite module 121 Fixed point 172 Form of type (p, q) 298 Frame 177 313 Frechet space 285 Frechet topology 285 Free group operation 172 Fubini metric 319 Fundamental form 315 Gauss's lemma 117 General linear group 172 Genus 232 Genus of a Riemann surface 340 Geodesic coordinates 316 Geometric series 11 Germ 123 192 Globally generated 176 Gluing 170 Gluing of fiber bundles 173 Gluing of manifolds 170 Godeaux surface 225 Good topological space 189 Graph 161 Graph of a meromorphic function 242 Grassmannian 212 225 Greatest common divisor 117 Griffiths negative 341 Griffiths positive 342 Harmonic form 335 356 Harmonic function 52 Hartogs convex 48 99 Hartogs figure, Euclidean 25 Hartogs figure, general 43 99 Hartogs triangle 365 Hartogs, theorem of 307 Hausdorff space 153 Hensel's lemma 120 Hermitian form 3 261 321 Hermitian metric 314 hessian 67 Hodge manifold 346 Hodge metric 346 Hodge star operator 323 Holder continuous 367 Holomorphic form 303 Holomorphic function 13 96 257 Holomorphic function on a complex manifold 156 Holomorphic function on an analytic set 134 Holomorphic gradient 28 Holomorphic map 30 158 Holomorphically convex 75 100 251 Holomorphically convex hull 75 Holomorphically separable 251 Holomorphically spreadable 251 Homogeneous 9 Homogeneous coordinates 209 Hopf bundle 238 Hopf manifold 208 225 Hopf's a-process 237 Hyperbolic manifold 368 Hyperelliptic involution 234 Hyperelliptic surface 232 Hyperplane at infinity 210 Hyperplane bundle 220 221 Hyperquadric 219 Hypersurface 127 220 Hypersurface, analytic 160 Ideal 108 Identity theorem for analytic sets 142 Identity theorem for holomorphic functions 22 Identity theorem for meromorphic functions 197 Identity theorem for power series 21 Identity theorem on manifolds 156 Immersion 168 Implicit function theorem 34 Incidence set 237 Indeterminacy, set of 243 Inner product, Euclidean 4 Inner product, Hermitian 3 Integral class 346 Integral domain 113 Inverse mapping theorem 33 Irreducible analytic set 141 Irreducible component 131 Irreducible element 117 Isomorphism of complex manifolds 159 Isomorphism of vector bundles 177 Jacobian, complex 30 Jacobian, real 31 Kahler manifold 316 Kahler metric 316 Kernel form 370 Kobayashi metric 368 Kodaira's embedding theorem 344 Krull topology 110 Kugelsatz 44 307 Laplacian 334 Leray covering see "Acyclic covering" Levi condition 66 Levi convex 66 Levi convex, strictly 66 Levi form 57 263 Levi's extension theorem 197 Levi's problem 51 Lie group 171 Lifted bundle 180 Lifting of a differential form 304 Limit of a filter 101 Line bundle 175 Line bundle, associated to a hypersurface 200 Line element 314 Linear subspace 218 Liouville theorem of 22 Local @-algebra 192 Local homeomorphism 88 Local parametrization 39 41 Local potential 318 Local uniformization 231 Locally compact 153 Locally finite covering 153 Logarithmically convex 83 Maximal ideal 108 Maximum principle 22 Maximum principle for plurisubharmonic functions 264
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