Several Complex Variables with Connections to Algebraic Geometry and Lie Groups Book

	Preface
Ch. 1	Selected Problems in One Complex Variable	1
1.1	Preliminaries	2
1.2	A Simple Problem	2
1.3	Partitions of Unity	4
1.4	The Cauchy-Riemann Equations	7
1.5	The Proof of Proposition 1.2.2	10
1.6	The Mittag-Leffler and Weierstrass Theorems	12
1.7	Conclusions and Comments	16
Ch. 2	Holomorphic Functions of Several Variables	23
2.1	Cauchy's Formula and Power Series Expansions	23
2.2	Hartog's Theorem	26
2.3	The Cauchy-Riemann Equations	29
2.4	Convergence Theorems	29
2.5	Domains of Holomorphy	31
Ch. 3	Local Rings and Varieties	37
3.1	Rings of Germs of Holomorphic Functions	38
3.2	Hilbert's Basis Theorem	39
3.3	The Weierstrass Theorems	40
3.4	The Local Ring of Holomorphic Functions is Noetherian	44
3.5	Varieties	45
3.6	Irreducible Varieties	49
3.7	Implicit and Inverse Mapping Theorems	50
3.8	Holomorphic Functions on a Subvariety	55
Ch. 4	The Nullstellensatz	61
4.1	Reduction to the Case of Prime Ideals	61
4.2	Survey of Results on Ring and Field Extensions	62
4.3	Hilbert's Nullstellensatz	68
4.4	Finite Branched Holomorphic Covers	72
4.5	The Nullstellensatz	79
4.6	Morphisms of Germs of Varieties	87
Ch. 5	Dimension	95
5.1	Topological Dimension	95
5.2	Subvarieties of Codimension 1	97
5.3	Krull Dimension	99
5.4	Tangential Dimension	100
5.5	Dimension and Regularity	103
5.6	Dimension of Algebraic Varieties	104
5.7	Algebraic vs. Holomorphic Dimension	108
Ch. 6	Homological Algebra	113
6.1	Abelian Categories	113
6.2	Complexes	119
6.3	Injective and Projective Resolutions	122
6.4	Higher Derived Functors	126
6.5	Ext	131
6.6	The Category of Modules, Tor	133
6.7	Hilbert's Syzygy Theorem	137
Ch. 7	Sheaves and Sheaf Cohomology	145
7.1	Sheaves	145
7.2	Morphisms of Sheaves	150
7.3	Operations on Sheaves	152
7.4	Sheaf Cohomology	157
7.5	Classes of Acyclic Sheaves	163
7.6	Ringed Spaces	168
7.7	De Rham Cohomology	172
7.8	Cech Cohomology	174
7.9	Line Bundles and Cech Cohomology	180
Ch. 8	Coherent Algebraic Sheaves	185
8.1	Abstract Varieties	186
8.2	Localization	189
8.3	Coherent and Quasi-coherent Algebraic Sheaves	194
8.4	Theorems of Artin-Rees and Krull	197
8.5	The Vanishing Theorem for Quasi-coherent Sheaves	199
8.6	Cohomological Characterization of Affine Varieties	200
8.7	Morphisms - Direct and Inverse Image	204
8.8	An Open Mapping Theorem	207
Ch. 9	Coherent Analytic Sheaves	215
9.1	Coherence in the Analytic Case	215
9.2	Oka's Theorem	217
9.3	Ideal Sheaves	221
9.4	Coherent Sheaves on Varieties	225
9.5	Morphisms between Coherent Sheaves	226
9.6	Direct and Inverse Image	229
Ch. 10	Stein Spaces	237
10.1	Dolbeault Cohomology	237
10.2	Chains of Syzygies	243
10.3	Functional Analysis Preliminaries	245
10.4	Cartan's Factorization Lemma	248
10.5	Amalgamation of Syzygies	252
10.6	Stein Spaces	257
Ch. 11	Frechet Sheaves - Cartan's Theorems	263
11.1	Topological Vector Spaces	264
11.2	The Topology of H(X)	266
11.3	Frechet Sheaves	274
11.4	Cartan's Theorems	277
11.5	Applications of Cartan's Theorems	281
11.6	Invertible Groups and Line Bundles	283
11.7	Meromorphic Functions	284
11.8	Holomorphic Functional Calculus	288
11.9	Localization	298
11.10	Coherent Sheaves on Compact Varieties	300
11.11	Schwartz's Theorem	302
Ch. 12	Projective Varieties	313
12.1	Complex Projective Space	313
12.2	Projective Space as an Algebraic and a Holomorphic Variety	314
12.3	The Sheaves O(k) and H(k)	317
12.4	Applications of the Sheaves O(k)	323
12.5	Embeddings in Projective Space	325
Ch. 13	Algebraic vs. Analytic - Serre's Theorems	331
13.1	Faithfully Flat Ring Extensions	331
13.2	Completion of Local Rings	334
13.3	Local Rings of Algebraic vs. Holomorphic Functions	338
13.4	The Algebraic to Holomorphic Functor	341
13.5	Serre's Theorems	344
13.6	Applications	351
Ch. 14	Lie Groups and Their Representations	357
14.1	Topological Groups	358
14.2	Compact Topological Groups	363
14.3	Lie Groups and Lie Algebras	376
14.4	Lie Algebras	385
14.5	Structure of Semisimple Lie Algebras	392
14.6	Representations of [actual symbol not reproducible][subscript 2]([Complex number system])	400
14.7	Representations of Semisimple Lie Algebras	404
14.8	Compact Semisimple Groups	409
Ch. 15	Algebraic Groups	419
15.1	Algebraic Groups and Their Representations	419
15.2	Quotients and Group Actions	423
15.3	Existence of the Quotient	427
15.4	Jordan Decomposition	430
15.5	Tori	433
15.6	Solvable Algebraic Groups	437
15.7	Semisimple Groups and Borel Subgroups	442
15.8	Complex Semisimple Lie Groups	451
Ch. 16	The Borel-Weil-Bott Theorem	459
16.1	Vector Bundles and Induced Representations	460
16.2	Equivariant Line Bundles on the Flag Variety	464
16.3	The Casimir Operator	469
16.4	The Borel-Weil Theorem	474
16.5	The Borel-Weil-Bott Theorem	478
16.6	Consequences for Real Semisimple Lie Groups	483
16.7	Infinite Dimensional Representations	484
	Bibliography	497
	Index	501