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CHAPTER 0 FOUNDATIONAL MATERIAL 1. Rudiments of Several Complex Variables Cauchy's Formula and Applications Several Variables Weierstrass Theorems and Corollaries Analytic Varieties 2. Complex Manifolds Complex Manifolds Submanifolds and Subvarieties De Rham and Dolbeault Cohomology Calculus on Complex Manifolds 3. Sheaves and Cohomology Origins: The Mittag-Leffler Problem Sheaves Cohomology of Sheaves The de Rham Theorem The Dolbeault Theorem 4. Topology of Manifolds Intersection of Cycles Poincare Duality Intersection of Analytic Cycles 5. Vector Bundles, Connections, and Curvature Complex and Holomorphic Vector Bundles Metrics, Connections, and Curvature 6. Harmonic Theory on Compact Complex Manifolds The Hodge Theorem Proof of the Hodge Theorem I: Local Theory Proof of the Hodge Theorem II: Global Theory Applications of the Hodge Theorem 7. Kahler Manifolds The Kahler Condition The Hodge Identities and the Hodge Decomposition The Lefschetz Decomposition CHAPTER 1 COMPLEX ALGEBRAIC VARIETIES 1. Divisors and Line Bundles Divisors Line Bundles Chern Classes of Line Bundles 2. Some Vanishing Theorems and Corollaries The Kodaira Vanishing Theorem The Lefschetz Theorem on Hyperplane Sections Theorem B The Lefschetz Theorem on (1, l)-classes 3. Algebraic Varieties Analytic and Algebraic Varieties Degree of a Variety Tangent Spaces to Algebraic Varieties 4. The Kodaira Embedding Theorem Line Bundles and Maps to Projective Space Blowing Up Proof of the Kodaira Theorem 5. Grassmannians Definitions The Cell Decomposition The Schubert Calculus Universal Bundles The Pliicker Embedding CHAPTER 2 RIEMANN SURFACES AND ALGEBRAIC CURVES Preliminaries Embedding Riemann Surfaces The Riemann-Hurwitz Formula The Genus Formula Cases g=0, 1 2. Abel's Theorem Abel's Theorem--First Version The First Reciprocity Law and Corollaries Abel's Theorem--Second Version Jacobi Inversion 3. Linear Systems on Curves Reciprocity Law II The Riemann-Roch Formula Canonical Curves Special Linear Systems I Hyperelliptic Curves and Riemann's Count Special Linear Systems II 4. Plucker Formulas Associated Curves Ramification The General Plucker Formulas I The General Plucker Formulas II Weierstrass Points Plucker Formulas for Plane Curves 5. Correspondences Definitions and Formulas Geometry of Space Curves Special Linear Systems III 6. Complex Tori and Abelian Varieties The Riemann Conditions Line Bundles on Complex Tori Theta-Functions The Group Structure on an Abelian Variety Intrinsic Formulations 7. Curves and Their Jacobians Preliminaries Riemann's Theorem Riemann's Singularity Theorem Special Linear Systems IV Torelli's Theorem CHAPTER 3 FURTHER TECHNIQUES 1. Distributions and Currents Definitions; Residue Formulas Smoothing and Regularity Cohomology of Currents 2. Applications of Currents to Complex Analysis Currents Associated to Analytic Varieties Intersection Numbers of Analytic Varieties The Levi Extension and Proper Mapping Theorems 3. Chern Classes Definitions The Gauss Bonnet Formulas Some Remarks--Not Indispensable--Concerning Chern Classes of Holomorphic Vector Bundles 4. Fixed-Point and Residue Formulas The Lefschetz Fixed-Point Formula The Holomorphic Lefschetz Fixed-Point Formula The Bott Residue Formula The General Hirzebruch-Riemann-Roch Formula 5. Spectral Sequences and Applications Spectral Sequences of Filtered and Bigraded Complexes Hypercohomology Differentials of the Second Kind The Leray Spectral Sequence CHAPTER 4 SURFACES 1. Preliminaries Intersection Numbers, the Adjunction Formula, and Riemann-Roch Blowing Up and Down The Quadric Surface The Cubic Surface 2. Rational Maps Rational and Birationai Maps Curves on an Algebraic Surface The Structure of Birational Maps Between Surfaces 3. Rational Surfaces I Noether's Lemma Rational Ruled Surfaces The General Rational Surface Surfaces of Minimal Degree Curves of Maximal Genus Steiner Constructions The Enriques-Petri Theorem 4. Rational Surfaces II The Castelnuovo-Enriques Theorem The Enriques Surface Cubic Surfaces Revisited The Intersection of Two Quadrics in p4 5. Some Irrational Surfaces The Albanese Map Irrational Ruled Surfaces A Brief Introduction to Elliptic Surfaces Kodaira Number and the Classification Theorem I The Classification Theorem II K-3 Surfaces Enriques Surfaces 6. Noether's Formula Noether's Formula for Smooth Hypersurfaces Blowing Up Submanifolds Ordinary Singularities of Surfaces Noether's Formula for General Surfaces Some Examples Isolated Singularities of Surfaces CHAPTER 5 RESIDUES 1. Elementary Properties of Residues Definition and Cohomological Interpretation The Global Residue Theorem The Transformation Law and Local Duality 2. Applications of Residues Intersection Numbers Finite Holomorphic Mappings Applications to Plane Projective Geometry 3. Rudiments of Commutative and Homological Algebra with Applications Commutative Algebra Homological Algebra The Koszul Complex and Applications A Brief Tour Through Coherent Sheaves 4. Global Duality Global Ext Explanation of the General Global Duality Theorem Global Ext and Vector Fields with Isolated Zeros Global Duality and Superabundance of Points on a Surface Extensions of Modules Points on a Surface and Rank-Two Vector Bundles Residues and Vector Bundles CHAPTER 6 THE QUADRIC LINE COMPLEX 1. Preliminaries: Quadrics Rank of a Quadric Linear Spaces on Quadrics Linear Systems of Quadrics Lines on Linear Systems of Quadrics The Problem of Five Conics 2. The Quadric Line Complex: Introduction Geometry of the Grassmannian G(2,4) Line Complexes The Quadric Line Complex and Associated Kummer Surface I Singular Lines of the Quadric Line Complex Two Configurations 3. Lines on the Quadric Line Complex The Variety of Lines on the Quadric Line Complex Curves on the Variety of Lines Two Configurations Revisited The Group Law 4. The Quadric Line Complex: Reprise The Quadric Line Complex and Associated Kummer Surface II Rationality of the Quadric Line Complex INDEX