Analytic and algebraic geometry : common problems, different methods

Analytic And Algebraic Geometers Often Study The Same Geometric Structures But Bring Different Methods To Bear On Them. While This Dual Approach Has Been Spectacularly Successful At Solving Problems, The Language Differences Between Algebra And Analysis Also Represent A Difficulty For Students And Researchers In Geometry, Particularly Complex Geometry. The Pcmi Program Was Designed To Partially Address This Language Gulf, By Presenting Some Of The Active Developments In Algebraic And Analytic Geometry In A Form Suitable For Students On The 'other Side' Of The Analysis-algebra Language Divide. One Focal Point Of The Summer School Was Multiplier Ideals, A Subject Of Wide Current Interest In Both Subjects. The Present Volume Is Based On A Series Of Lectures At The Pcmi Summer School On Analytic And Algebraic Geometry. The Series Is Designed To Give A High-level Introduction To The Advanced Techniques Behind Some Recent Developments In Algebraic And Analytic Geometry. The Lectures Contain Many Illustrative Examples, Detailed Computations, And New Perspectives On The Topics Presented, In Order To Enhance Access Of This Material To Non-specialists.--publisher's Description. Machine Generated Contents Note: An Introduction To Things & Part; / Bo Berndtsson -- Introduction -- Lecture 1 The One-dimensional Case -- 1.1. The & Part;-equation In One Variable -- 1.2. An Alternative Proof Of The Basic Identity -- 1.3. An Application: Inequalities Of Brunn-minkowski Type -- 1.4. Regularity -- A Disclaimer -- Lecture 2 Functional Analytic Interlude -- 2.1. Dual Formulation Of The & Part;-problem -- Lecture 3 The & Part;-equation On A Complex Manifold -- 3.1. Metrics -- 3.2. Norms Of Forms -- 3.3. Line Bundles -- 3.4. Calculation Of The Adjoint And The Basic Identity -- 3.5. The Main Existence Theorem And L2-estimate For Compact Manifolds -- 3.6. Complete Kahler Manifolds -- Lecture 4 The Bergman Kernel -- 4.1. Generalities -- 4.2. Bergman Kernels Associated To Complex Line Bundles -- Lecture 5 Singular Metrics And The Kawamata-viehweg Vanishing Theorem -- 5.1. The Demailly-nadel Vanishing Theorem -- 5.2. The Kodaira Embedding Theorem. 5.3. The Kawamata-viehweg Vanishing Theorem -- Lecture 6 Adjunction And Extension From Divisors -- 6.1. Adjunction And The Currents Defined By Divisors -- 6.2. The Ohsawa-takegoshi Extension Theorem -- Lecture 7 Deformational Invariance Of Plurigenera -- 7.1. Extension Of Pluricanonical Forms -- Bibliography -- Real And Complex Geometry Meet The Cauchy-riemann Equations / John P. D'angelo -- Preface -- Lecture 1 Background Material -- 1. Complex Linear Algebra -- 2. Differential Forms -- 3. Solving The Cauchy-riemann Equations -- Lecture 2 Complex Varieties In Real Hypersurfaces -- 1. Degenerate Critical Points Of Smooth Functions -- 2. Hermitian Symmetry And Polarization -- 3. Holomorphic Decomposition -- 4. Real Analytic Hypersurfaces And Subvarieties -- 5. Complex Varieties, Local Algebra, And Multiplicities -- Lecture 3 Pseudoconvexity, The Levi Form, And Points Of Finite Type -- 1. Euclidean Convexity -- 2. The Levi Form -- 3. Higher Order Commutators -- 4. Points Of Finite Type -- 5. Commutative Algebra. 6. A Return To Finite Type -- 7. The Set Of Finite Type Points Is Open -- Lecture 4 Kohn's Algorithm For Subelliptic Multipliers -- 1. Introduction -- 2. Subelliptic Estimates -- 3. Kohn's Algorithm -- 4. Kohn's Algorithm For Holomorphic And Formal Germs -- 5. Failure Of Effectiveness For Kohn's Algorithm -- 6. Triangular Systems -- 7. Additional Remarks -- Lecture 5 Connections With Partial Differential Equations -- 1. Finite Type Conditions -- 2. Local Regularity For & Part; -- 3. Hypoellipticity, Global Regularity, And Compactness -- 4. An Introduction To L2-estimates -- Lecture 6 Positivity Conditions -- 1. Introduction -- 2. The Classes P & Kappa; -- 3. Intermediate Conditions -- 4. The Global Cauchy-schwarz Inequality -- 5. A Complicated Example -- 6. Stabilization In The Bihomogeneous Polynomial Case -- 7. Squared Norms And Proper Mappings Between Balls -- 8. Holomorphic Line Bundles -- Lecture 7 Some Open Problems -- Bibliography -- Three Variations On A Theme In Complex Analytic Geometry / Dror Varolin. Lecture 0 Basic Notions In Complex Geometry -- 1. Complex Manifolds -- 2. Connections -- 3. Curvature -- 4. Holomorphic Line Bundles -- Lecture 1 The Hormander Theorem -- 1. Functional Analysis -- 2. The Bochner-kodaira Identity -- 3. Manifolds With Boundary -- 4. Density Of Smooth Forms In The Graph Norm -- 5. Hormander's Theorem -- 6. Singular Hermitian Metrics For Line Bundles -- 7. Application: Kodaira Embedding Theorem -- 8. Multiplier Ideal Sheaves And Nadel's Theorems -- 9. Exercises -- Lecture 2 The L2 Extension Theorem -- 1. L2 Extension -- 2. The Deformation Invariance Of Plurigenera -- 3. Pluricanonical Extension On Projective Manifolds -- 4. Exercises -- Lecture 3 The Skoda Division Theorem -- 1. Statement Of The Division Theorem -- 2. Proof Of The Division Theorem -- 3. Global Generation Of Multiplier Ideal Sheaves -- 4. Exercises -- Bibliography -- Structure Theorems For Projective And Kahler Varieties / Jean-pierre Demailly -- 0. Introduction -- 1. Numerically Effective And Pseudo-effective (1,1) Classes. 1.a. Pseudo-effective Line Bundles And Metrics With Minimal Singularities -- 1.b. Nef Line Bundles -- 1.c. Description Of The Positive Cones -- 1.d. The Kawamata-viehweg Vanishing Theorem -- 1.e. A Uniform Global Generation Property Due To Y.t. Siu -- 1.f. Hard Lefschetz Theorem With Multiplier Ideal Sheaves -- 2. Holomorphic Morse Inequalities -- 3. Approximation Of Closed Positive (1,1)-currents By Divisors -- 3.a. Local Approximation Theorem Through Bergman Kernels -- 3.b. Global Approximation Of Closed (1,1)-currents On A Compact Complex Manifold -- 3.c. Global Approximation By Divisors -- 3.d. Singularity Exponents And Log Canonical Thresholds -- 4. Subadditivity Of Multiplier Ideals And Fujita's Approximate Zariski Decomposition Theorem -- 5. Numerical Characterization Of The Kahler Cone -- 5.a. Positive Classes In Intermediate (p, P) Bidegrees -- 5.b. Numerically Positive Classes Of Type (1,1) -- 5.c. Deformations Of Compact Kahler Manifolds -- 6. Structure Of The Pseudo-effective Cone And Mobile Intersection Theory. 6.a. Classes Of Mobile Curves And Of Mobile (n -- 1, N -- 1)-currents -- 6.b. Zariski Decomposition And Mobile Intersections -- 6.c. The Orthogonality Estimate -- 6.d. Dual Of The Pseudo-effective Cone -- 7. Super-canonical Metrics And Abundance -- 7.a. Construction Of Super-canonical Metrics -- 7.b. Invariance Of Plurigenera And Positivity Of Curvature Of Super-canonical Metrics -- 7.c. Tsuji's Strategy For Studying Abundance -- 8. Siu's Analytic Approach And Paun's Non Vanishing Theorem -- Bibliography -- Lecture Notes On Rational Polytopes And Finite Generation / Mihai Paun -- 0. Introduction -- 1. Basic Definitions And Notations -- 2. Proof Of (i) -- 2.1. The Case Nd({kx + Yto + A}) = 0 -- 2.2. The X Method For Sequences -- 2.3. The Induced Polytope And Its Properties -- 3. Proof Of (ii) -- 3.1. The First Step -- 3.2. Iteration Scheme -- References -- Introduction To Resolution Of Singularities / Mircea Mustata -- Lecture 1 Resolutions And Principalizations -- 1.1. The Main Theorems. 1.2. Strengthenings Of Theorem 1.3 -- 1.3. Historical Comments -- Lecture 2 Marked Ideals -- 2.1. Marked Ideals -- 2.2. Derived Ideals -- Lecture 3 Hypersurfaces Of Maximal Contact And Coefficient Ideals -- 3.1. Hypersurfaces Of Maximal Contact -- 3.2. The Coefficient Ideal -- Lecture 4 Homogenized Ideals -- 4.1. Basics Of Homogenized Ideals -- 4.2. Comparing Hypersurfaces Of Maximal Contact: Formal Equivalence -- 4.3. Comparing Hypersurfaces Of Maximal Contact: Etale Equivalence -- Lecture 5 Proof Of Principalization -- 5.1. The Statements -- 5.2. Part I: The Maximal Order Case -- 5.3. Part Ii: The General Case -- 5.4. Proof Of Principalization -- Bibliography -- A Short Course On Multiplier Ideals / Robert Lazarsfeld -- Introduction -- Lecture 1 Construction And Examples Of Multiplier Ideals -- Definition Of Multiplier Ideals -- Monomial Ideals -- Invariants Defined By Multiplier Ideals -- Lecture 2 Vanishing Theorems For Multiplier Ideals -- The Kawamata-viehweg-nadel Vanishing Theorem -- Singularities Of Plane Curves And Projective Hypersurfaces -- Singularities Of Theta Divisors -- Uniform Global Generation. Lecture 3 Local Properties Of Multiplier Ideals -- Adjoint Ideals And The Restriction Theorem -- The Subadditivity Theorem -- Skoda's Theorem -- Lecture 4 Asymptotic Constructions -- Asymptotic Multiplier Ideals -- Variants -- Etale Multiplicativity Of Plurigenera -- A Comparison Theorem For Symbolic Powers -- Lecture 5 Extension Theorems And Deformation Invariance Of Plurigenera -- Bibliography -- Exercises In The Birational Geometry Of Algebraic Varieties / Janos Kollar -- 1. Birational Classification Of Algebraic Surfaces -- 2. Naive Minimal Models -- 3. The Cone Of Curves -- 4. Singularities -- 5. Flips -- 6. Minimal Models -- Bibliography -- Higher Dimensional Minimal Model Program For Varieties Of Log General Type / Christopher D. Hacon -- Introduction -- Lecture 1 Pl-flips -- Lecture 2 Multiplier Ideal Sheaves -- Asymptotic Multiplier Ideal Sheaves -- Extending Pluricanonical Forms -- 3. Finite Generation Of The Restricted Algebra -- Rationality Of The Restricted Algebra -- Proof Of (1.10) -- Lecture 4 The Minimal Model Program With Scaling -- Solutions To The Exercises -- Bibliography -- Lectures On Flips And Minimal Models / Mircea Mustata. Lecture 1 Extension Theorems -- 1.1. Multiplier And Adjoint Ideals -- 1.2. Proof Of The Main Lemma -- Lecture 2 Existence Of Flips I -- 2.1. The Setup -- 2.2. Adjoint Algebras -- 2.3. The Hacon-mckernan Extension Theorem -- 2.4. The Restricted Algebra As An Adjoint Algebra -- Lecture 3 Existence Of Flips Ii -- Lecture 4 Notes On Birkar-cascini-hacon-mckernan -- 4.1. Comparison Of 3 Mmp's -- 4.2. Mmp With Scaling -- 4.3. Mmp With Scaling Near [& Delta;] -- 4.4. Bending It Like Bchm -- 4.5. Finiteness Of Models. Jeffery Mcneal, Mircea Mustata, Editors. Includes Bibliographical References (p. 583)