Preface
Jeffery McNeal and Mircea Mustata
Introduction
Bo Berndtsson
An Introduction to Things
Introduction
Lecture 1. The one-dimensional case
1.1. The 0-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 problem
Lecture 3. The -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
John P. D'Angelo
Real and Complex Geometry meet the Cauchy-Riemann Equations
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
3. Hypoellipticity, global regularity, and compactness
4. An introduction to L2-estimates
Lecture 6. Positivity conditions
1. Introduction
2. The classes Pk
3. Intermediate conditions
4. The global Canchy-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
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