From Holomorphic Functions to Complex Manifolds

by Fritzsche, Klaus; Grauert, Hans

• ISBN13: 9780387953953
• ISBN10: 0387953957
• Format: Hardcover
• Copyright: 2002-04-01
• Publisher: Springer Nature

Summary

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. The book can be used as a first introduction to several complex variables as well as a reference for the expert.

Author Biography

Hans Grauert is a professor at the University of Gottingen.

Table of Contents

Preface	v
Holomorphic Functions	1	(42)
Complex Geometry	1	(8)
Real and Complex Structures	1	(2)
Hermitian Forms and Inner Products	3	(2)
Balls and Polydisks	5	(1)
Connectedness	6	(1)
Reinhardt Domains	7	(2)
Power Series	9	(5)
Polynomials	9	(1)
Convergence	9	(2)
Power Series	11	(3)
Complex Differentiable Functions	14	(3)
The Complex Gradient	14	(1)
Weakly Holomorphic Functions	15	(1)
Holomorphic Functions	16	(1)
The Cauchy Integral	17	(6)
The Integral Formula	17	(2)
Holomorphy of the Derivatives	19	(3)
The Identity Theorem	22	(1)
The Hartogs Figure	23	(3)
Expansion in Reinhardt Domains	23	(2)
Hartogs Figures	25	(1)
The Cauchy-Riemann Equations	26	(4)
Real Differentiable Functions	26	(2)
Wirtinger's Calculus	28	(1)
The Cauchy-Riemann Equations	29	(1)
Holomorphic Maps	30	(6)
The Jacobian	30	(2)
Chain Rules	32	(1)
Tangent Vectors	32	(1)
The Inverse Mapping	33	(3)
Analytic Sets	36	(7)
Analytic Subsets	36	(2)
Bounded Holomorphic Functions	38	(1)
Regular Points	39	(2)
Injective Holomorphic Mappings	41	(2)
Domains of Holomorphy	43	(62)
The Continuity Theorem	43	(9)
General Hartogs Figures	43	(2)
Removable Singularities	45	(2)
The Continuity Principle	47	(1)
Hartogs Convexity	48	(1)
Domains of Holomorphy	49	(3)
Plurisubharmonic Functions	52	(8)
Subharmonic Functions	52	(3)
The Maximum Principle	55	(1)
Differentiable Subharmonic Functions	55	(1)
Plurisubharmonic Functions	56	(1)
The Levi Form	57	(1)
Exhaustion Functions	58	(2)
Pseudoconvexity	60	(4)
Pseudoconvexity	60	(1)
The Boundary Distance	60	(3)
Properties of Pseudoconvex Domains	63	(1)
Levi Convex Boundaries	64	(9)
Boundary Functions	64	(2)
The Levi Condition	66	(1)
Affine Convexity	66	(3)
A Theorem of Levi	69	(4)
Holomorphic Convexity	73	(5)
Affine Convexity	73	(2)
Holomorphic Convexity	75	(1)
The Cartan-Thullen Theorem	76	(2)
Singular Functions	78	(4)
Normal Exhaustions	78	(1)
Unbounded Holomorphic Functions	79	(1)
Sequences	80	(2)
Examples and Applications	82	(5)
Domains of Holomorphy	82	(1)
Complete Reinhardt Domains	83	(2)
Analytic Polyhedra	85	(2)
Riemann Domains over Cn	87	(9)
Riemann Domains	87	(4)
Union of Riemann Domains	91	(5)
The Envelope of Holomorphy	96	(9)
Holomorphy on Riemann Domains	96	(1)
Envelopes of Holomorphy	97	(2)
Pseudoconvexity	99	(1)
Boundary Points	100	(2)
Analytic Disks	102	(3)
Analytic Sets	105	(48)
The Algebra of Power Series	105	(5)
The Banach Algebra Bt	105	(1)
Expansion with Respect to z1	106	(1)
Convergent Series in Banach Algebras	107	(1)
Convergent Power Series	108	(1)
Distinguished Directions	109	(1)
The Preparation Theorem	110	(6)
Division with Remainder in Bt	110	(3)
The Weierstrass Condition	113	(1)
Weierstrass Polynomials	114	(1)
Weierstrass Preparation Theorem	115	(1)
Prime Factorization	116	(7)
Unique Factorization	116	(1)
Gauss's Lemma	117	(2)
Factorization in Hn	119	(1)
Hensel's Lemma	119	(1)
The Noetherian Property	120	(3)
Branched Coverings	123	(12)
Germs	123	(1)
Pseudopolynomials	124	(1)
Euclidean Domains	125	(1)
The Algebraic Derivative	125	(1)
Symmetric Polynomials	126	(1)
The Discriminant	126	(1)
Hypersurfaces	127	(3)
The Unbranched Part	130	(1)
Decompositions	130	(2)
Projections	132	(3)
Irreducible Components	135	(8)
Embedded-Analytic Sets	135	(2)
Images of Embedded-Analytic Sets	137	(1)
Local Decomposition	138	(2)
Analyticity	140	(1)
The Zariski Topology	141	(1)
Global Decompositions	141	(2)
Regular and Singular Points	143	(10)
Compact Analytic Sets	143	(1)
Embedding of Analytic Sets	144	(1)
Regular Points of an Analytic Set	145	(2)
The Singular Locus	147	(1)
Extending Analytic Sets	147	(3)
The Local Dimension	150	(3)
Complex Manifolds	153	(98)
The Complex Structure	153	(18)
Complex Coordinates	153	(3)
Holomorphic Functions	156	(1)
Riemann Surfaces	157	(1)
Holomorphic Mappings	158	(1)
Cartesian Products	159	(1)
Analytic Subsets	160	(2)
Differentiable Functions	162	(2)
Tangent Vectors	164	(2)
The Complex Structure on the Space of Derivations	166	(1)
The Induced Mapping	167	(1)
Immersions and Submersions	168	(2)
Gluing	170	(1)
Complex Fiber Bundles	171	(11)
Lie Groups and Transformation Groups	171	(2)
Fiber Bundles	173	(1)
Equivalence	174	(1)
Complex Vector Bundles	175	(2)
Standard Constructions	177	(3)
Lifting of Bundles	180	(1)
Subbundles and Quotients	180	(2)
Cohomology	182	(10)
Cohomology Groups	182	(2)
Refinements	184	(1)
Acyclic Coverings	185	(1)
Generalizations	186	(2)
The Singular Cohomology	188	(4)
Meromorphic Functions and Divisors	192	(11)
The Ring of Germs	192	(1)
Analytic Hypersurfaces	193	(3)
Meromorphic Functions	196	(2)
Divisors	198	(2)
Associated Line Bundles	200	(1)
Meromorphic Sections	201	(2)
Quotients and Submanifolds	203	(23)
Topological Quotients	203	(1)
Analytic Decompositions	204	(1)
Properly Discontinuously Acting Groups	205	(1)
Complex Tori	206	(1)
Hopf Manifolds	207	(1)
The Complex Projective Space	208	(2)
Meromorphic Functions	210	(1)
Grassmannian Manifolds	211	(3)
Submanifolds and Normal Bundles	214	(2)
Projective Algebraic Manifolds	216	(3)
Projective Hypersurfaces	219	(3)
The Euler Sequence	222	(1)
Rational Functions	223	(3)
Branched Riemann Domains	226	(9)
Branched Analytic Coverings	226	(2)
Branched Domains	228	(1)
Torsion Points	229	(1)
Concrete Riemann Surfaces	230	(1)
Hyperelliptic Riemann Surfaces	231	(4)
Modifications and Toric Closures	235	(16)
Proper Modifications	235	(2)
Blowing Up	237	(1)
The Tautological Bundle	237	(4)
Quadratic Transformations	241	(1)
Monoidal Transformations	241	(1)
Meromorphic Maps	242	(2)
Toric Closures	244	(7)
Stein Theory	251	(46)
Stein Manifolds	251	(9)
Introduction	251	(1)
Fundamental Theorems	252	(1)
Cousin-I Distributions	253	(1)
Cousin-II Distributions	254	(1)
Chern Class and Exponential Sequence	255	(2)
Extension from Submanifolds	257	(1)
Unbranched Domains of Holomorphy	257	(1)
The Embedding Theorem	258	(1)
The Serre Problem	259	(1)
The Levi Form	260	(6)
Covariant Tangent Vectors	260	(1)
Hermitian Forms	261	(1)
Coordinate Transformations	262	(1)
Plurisubharmonic Functions	263	(1)
The Maximum Principle	264	(2)
Pseudoconvexity	266	(10)
Pseudoconvex Complex Manifolds	266	(1)
Examples	267	(7)
Analytic Tangents	274	(2)
Cuboids	276	(6)
Distinguished Cuboids	276	(1)
Vanishing of Cohomology	277	(1)
Vanishing on the Embedded Manifolds	278	(1)
Cuboids in a Complex Manifold	278	(2)
Enlarging U'	280	(1)
Approximation	281	(1)
Special Coverings	282	(7)
Cuboid Coverings	282	(1)
The Bubble Method	283	(1)
Frechet Spaces	284	(2)
Finiteness of Cohomology	286	(1)
Holomorphic Convexity	286	(1)
Negative Line Bundles	287	(1)
Bundles over Stein Manifolds	288	(1)
The Levi Problem	289	(8)
Enlarging: The Idea of the Proof	289	(1)
Enlarging: The First Step	290	(2)
Enlarging: The Whole Process	292	(1)
Solution of the Levi Problem	293	(2)
The Compact Case	295	(2)
Kahler Manifolds	297	(58)
Differential Forms	297	(6)
The Exterior Algebra	297	(1)
Forms of Type (p, q)	298	(2)
Bundles of Differential Forms	300	(3)
Dolbeault Theory	303	(11)
Integration of Differential Forms	303	(2)
The Inhomogeneous Cauchy Formula	305	(1)
The ∂-Equation in One Variable	306	(1)
A Theorem of Hartogs	307	(1)
Dolbeault's Lemma	308	(2)
Dolbeault Groups	310	(4)
Kahler Metrics	314	(8)
Hermitian metrics	314	(1)
The Fundamental Form	315	(1)
Geodesic Coordinates	316	(1)
Local Potentials	317	(1)
Pluriharmonic Functions	318	(1)
The Fubini Metric	318	(2)
Deformations	320	(2)
The Inner Product	322	(7)
The Volume Element	322	(1)
The Star Operator	323	(1)
The Effect on (p, q)-Forms	324	(3)
The Global Inner Product	327	(1)
Currents	328	(1)
Hodge Decomposition	329	(12)
Adjoint Operators	329	(2)
The Kahlerian Case	331	(1)
Bracket Relations	332	(2)
The Laplacian	334	(1)
Harmonic Forms	335	(3)
Consequences	338	(3)
Hodge Manifolds	341	(7)
Negative Line Bundles	341	(1)
Special Holomorphic Cross Sections	342	(2)
Projective Embeddings	344	(1)
Hodge Metrics	345	(3)
Applications	348	(7)
Period Relations	348	(4)
The Siegel Upper Halfplane	352	(1)
Semipositive Line Bundles	352	(1)
Moishezon Manifolds	353	(2)
Boundary Behavior	355	(20)
Strongly Pseudoconvex Manifolds	355	(2)
The Hilbert Space	355	(1)
Operators	355	(2)
Boundary Conditions	357	(1)
Subelliptic Estimates	357	(7)
Sobolev Spaces	357	(2)
The Neumann Operator	359	(1)
Real-Analytic Boundaries	360	(1)
Examples	360	(4)
Nebenhullen	364	(3)
General Domains	364	(1)
A Domain with Nontrivial Nebenhulle	365	(1)
Bounded Domains	366	(1)
Domains in C2	366	(1)
Boundary Behavior of Biholomorphic Maps	367	(8)
The One-Dimensional Case	367	(1)
The Theory of Henkin and Vormoor	367	(2)
Real-Analytic Boundaries	369	(1)
Fefferman's Result	369	(2)
Mappings	371	(1)
The Bergman Metric	371	(4)
References	375	(6)
Index of Notation	381	(6)
Index	387