Complex Analysis: Lectures given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Bressanone (Bolzano), Italy, June 3-12, 1973 (C.I.M.E. Summer Schools (62))

Annotation A. Andreotti: Nine lectures on complex analysis.- J.J. Kohn: Propagations of singularities for the Cauchy-Riemann equations.- Yum-Tong Siu: The mixed case of the direct image theorem and its applications Cover Title Copyright Contents NINE LECTURES ON COMPLES ANALYSIS Preface CONTENTS Chapter 1. Elementary theory of holomorohic convexity 1.1 Preliminaries 1.2 Hartogs domains 1.3 Open seta of holomorphy 1.4 Levi (1)-convexity Exerclses Chapter II. Pseudoconcave manifolds 2.1 Preliminaries 2.2. Mermorohic functions and holomorphic line bundles 2.3 Pseudoconcave manifolds 2.4. Analytic and algebraic dspendence of meromorphic functions 2.5. Algebraic fields of meromorphic functions Chapter III. Properly discontinuous pseudoconcave groups:the Siegel modular group 3.1. Preliminaries 3.2. Pauedoconcave properlY discontinuous groups 3.3 Siegel modular group 3.4 Pseudoconcavity of the modular group 3.5. Poincare series Chapter IV. Projective imbeddings of Dseudoconcave manifolds 4.1. Measure of pseudoconcavity 4.2 The problem of projective imbedding of pseudoconcave manifolds 4.3 Solution of the problem for o-pseudoconcave .manifolds 4.4. The case of dimE X >=3. Chapter V:. Meromorphic functions on complex spaces 5.1 Preliminaries 5.2 Pseudoconcavity for complex spaces 5.3 The Poincare problem 5.4. Relative theorems Chapter VI: .E. E. Levi problem. 6.1 Preliminaries. 6.2 E.E. Levi problem 6.3 Proof of Grauert's theorem 6.4. Characterization of projective algebraic manifolds,Kodaira's theorem Chapter VII:. Generalizations of the Levi-problem 7.1. d-open sets of holomorphy 7.2. Proof of theorem (Grauert) 7.3 Finiteness theorems 7.4 Applications to projective algebraic manifolds Chapter VIII. Duality theorems on complex manifolds 8.1 Preliminaries. 8.2. Cech homology on complex manifolds 8.3. Duality between cohomology and homology 8.4. Cech homology and the functor EXT 8.5 Divisors and Riemann-Roch theorem Chapter IX. The H. Lewy problem 9.1. Preliminaries 9.2. Mayer-Vietrois sequence 9.3. Bochner theorem 9.4. Riemann-Hilbert and Cauchy problem 9.5. Cauchy-problem as a Vanishing theorem for cohomology 9.6. Non-validity of Poincare lemma for the complex BIBLIOGRAPHY PROPAGATION OF SINGULARITIESFOR THE CAUCHY. RIEMANN EQUATIONS Introduction Leoture 1. The a- problem and Hartog's theorem Lecture 2. Pseudo-convexity Lecture 3. Pormulation of the a-Neumann problem Lecture 4. The bastc a prtort esttmates Lecture 5. Pseudo-differential operators Lecture 6. Interlor regularlty and exlstence theorems Lecture 7. Boundary regularity Lecture 8. The 1nduced Cauchy-Rlemann equat10ns References THE MIXED CASE OF THE DIRECT IMAGETHEOREM AND ITS APPLICATIONS THB MIXED CASE OF THE DIRECT IMAGE THEOREMAND ITS APPLICATIONS § 0. Introduction Table of Contents PART I: CONSTRUCTION OF SlMPLEXES OF BANACH BUNDLES §l Privileged Polydiscs §2 Semi-norms on Unreduced Spaces §3.Theorem B with Bounds §4. Leray's Theorem with Bounds §5. Extension of Cohomology Classes §6. Sheaf Systems PART II THE POWER SERIES METHOD §7 Finite Generation with fuunds §8. Right Inverses of Coboundary Maps §9. Global Isomorphism §10. Proof of Coherence PART III: APPLICATIONS §11. Coherent Sheaf Extension §12. Blow-downs §13. Relative Exceptional Sets §14. Projectivity Criterion §15. Extension of Complex Spaces APPENDIX REFERENCES