PrefaceJeffery McNeal and Mircea MustataIntroductionBo BerndtssonAn Introduction to ThingsIntroductionLecture 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 disclaimerLecture 2. Functional analytic interlude 2.1. Dual formulation of the problemLecture 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 manifoldsLecture 4. The Bergman kernel 4.1. Generalities 4.2. Bergman kernels associated to complex line bundlesLecture 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 theoremLecture 6. Adjunction and extension from divisors 6.1. Adjunction and the currents defined by divisors 6.2. The Ohsawa-Takegoshi extension theoremLecture 7. Deformational invariance of plurigenera 7.1. Extension of pluricanonical formsBibliographyJohn P. D'AngeloReal and Complex Geometry meet the Cauchy-Riemann EquationsPrefaceLecture 1. Background material 1. Complex linear algebra 2. Differential forms 3. Solving the Cauchy-Riemann equationsLecture 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 multiplicitiesLecture 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 openLecture 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 remarksLecture 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-estimatesLecture 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����