Contents
PART I
1. Complex differentiation 3
1.1 Real and complex differentials 3
1.2 The a and [) differentials 5
1.3 Real and complex Jacobians 7
1.4 The Levi form and the second differential 8
1.5 Complex differential forms 12
Exercises 15
2. Subharmonic and plurisubharmonic functions 20
2.1 Integral averages 20
2.2 Harmonic functions 25
2.3 Semicontinuity 36
2.4 Subharmonic functions 38
2.5 Subharmonicity and smoothing 42
2.6 Families of subharmonic functions 49
2.7 Removable singularities of subharmonic functions 52
2.8 Applications to holomorphic functions 55
2.9 Plurisubharmonic functions 62
2.10 Pseudoconvexity 74
Exercises 81
PART II
3. The complex Monge-Ampere operator 87
3.1 Maximal plurisubharmonic functions 87
3.2 Positive alternating forms 100
3.3 Currents 104
3.4 The complex Monge-Ampere operator 110
3.5 Quasicontinuity of plurisubharmonic functions 120
3.6 Continuity properties of the Monge-Ampere operator 125
3.7 Comparison theorems 126
3.8 Discontinuity of the Monge-Ampere operator 131
4. The Dirichlet problem for the Monge-Ampere operator 134
4.1 The Riesz decomposition and applications 135
4.2 Generalized second order differential operators 139
4.3 Regularity of the Perron-Bremermann function 148
4.4 The Bedford-Taylor existence theorem155
4.5 The relative extremal functions 158
4.6 Negligible sets and the relative capacity 165
4.7 Applications to pluripolar sets 168
4.8 Pluri-thin and other small sets in en 173
5. Maximal functions of logarithmic growth 182
5.1 Pluricomplex Green functions with pole at infinity 184
5.2 C-polar sets 191
5.3 Invariance and criteria of L-regularity 195
5.4 Pluricomplex Green functions for subsets of Rn 203
5.5 Complex equilibrium measures 209
5.6 Equilibrium measures and families of polynomials 212
6. Maximal functions with logarithmic singulari ties 220
6.1 Pluricomplex Green function with a logarithmic pole 221
6.2 Continuity properties of the Green function 225
6.3 The Green function and the Monge-Ampere operator 228
6.4 Comparisons between the pluricomplex and the one-dimensional cases 232
6.5 Applications of the Green function 235
Appendix: Foliations 245
References 248
Index 263
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