complex geometry - jean-pierre demailly

Complex Analytic and Dierential GeometryJean-Pierre DemaillyUniversit de Grenoble I e Institut Fourier, UMR 5582 du CNRS 38402 Saint-Martin dH`res, France e

Version of Thursday September 10, 2009

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Table of Contents

Chapter I. Complex Dierential Calculus and Pseudoconvexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1. 2. 3. 4. 5. 6. 7. 8. Dierential Calculus on Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Currents on Dierentiable Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Holomorphic Functions and Complex Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Subharmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Plurisubharmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Domains of Holomorphy and Stein Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Pseudoconvex Open Sets in Cn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .60

Chapter II. Coherent Sheaves and Analytic Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 1. Presheaves and Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 2. The Local Ring of Germs of Analytic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3. Coherent Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4. Complex Analytic Sets. Local Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5. Complex Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6. Analytic Cycles and Meromorphic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 7. Normal Spaces and Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 8. Holomorphic Mappings and Extension Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 9. Complex Analytic Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 10. Bimeromorphic maps, Modications and Blow-ups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 11. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Chapter III. Positive Currents and Lelong Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 1. Basic Concepts of Positivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 2. Closed Positive Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 3. Denition of Monge-Amp`re Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 e 4. Case of Unbounded Plurisubharmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 5. Generalized Lelong Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 6. The Jensen-Lelong Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 7. Comparison Theorems for Lelong Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 8. Sius Semicontinuity Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 9. Transformation of Lelong Numbers by Direct Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 10. A Schwarz Lemma. Application to Number Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 Chapter IV. Sheaf Cohomology and Spectral Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 1. 2. 3. 4. Basic Results of Homological Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 The Simplicial Flabby Resolution of a Sheaf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 Cohomology Groups with Values in a Sheaf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Acyclic Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

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Table of Contents 5. Cech Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 6. The De Rham-Weil Isomorphism Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 7. Cohomology with Supports. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .216 8. Cup Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 9. Inverse Images and Cartesian Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 10. Spectral Sequence of a Filtered Complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 11. Spectral Sequence of a Double Complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 12. Hypercohomology Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 13. Direct Images and the Leray Spectral Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 14. Alexander-Spanier Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 15. Knneth Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 u 16. Poincar duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 e

Chapter V. Hermitian Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 1. Denition of Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 2. Linear Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 3. Curvature Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 4. Operations on Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 5. Pull-Back of a Vector Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 6. Parallel Translation and Flat Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 7. Hermitian Vector Bundles and Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 8. Vector Bundles and Locally Free Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 9. First Chern Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 10. Connections of Type (1,0) and (0,1) over Complex Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 11. Holomorphic Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 12. Chern Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 13. Lelong-Poincar Equation and First Chern Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 e 14. Exact Sequences of Hermitian Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 15. Line Bundles (k) over Pn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 16. Grassmannians and Universal Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 Chapter VI. Hodge Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 1. Dierential Operators on Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 2. Formalism of PseudoDierential Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 3. Harmonic Forms and Hodge Theory on Riemannian Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 4. Hermitian and Khler Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 a 5. Basic Results of Khler Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 a 6. Commutation Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 7. Groups p,q (X, E) and Serre Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 8. Cohomology of Compact Khler Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 a 9. Jacobian and Albanese Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 10. Complex Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 11. Hodge-Frlicher Spectral Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 o 12. Eect of a Modication on Hodge Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 Chapter VII. Positive Vector Bundles and Vanishing Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 329 1. 2. 3. 4. 5. 6. Bochner-Kodaira-Nakano Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 Basic a Priori Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 Kodaira-Akizuki-Nakano Vanishing Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 Girbaus Vanishing Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 Vanishing Theorem for Partially Positive Line Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 Positivity Concepts for Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338

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7. Nakano Vanishing Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 8. Relations Between Nakano and Griths Positivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 9. Applications to Griths Positive Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 10. Cohomology Groups of (k) over Pn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 11. Ample Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 12. Blowing-up along a Submanifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 13. Equivalence of Positivity and Ampleness for Line Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 14. Kodairas Projectivity Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 Chapter VIII. L2 Estimates on Pseudoconvex Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .363 1. Non Bounded Operators on Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 2. Complete Riemannian Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 3. L2 Hodge Theory on Complete Riemannian Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 4. General Estimate for d on Hermitian Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 5. Estimates on Weakly Pseudoconvex Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 6. Hrmanders Estimates for non Complete Khler Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 o a 7. Extension of Holomorphic Functions from Subvarieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 8. Applications to Hypersurface Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 9. Skodas L2 Estimates for Surjective Bundle Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 10. Application of Skodas L2 Estimates to Local Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 11. Integrability of Almost Complex Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 Chapter IX. Finiteness Theorems for q-Convex Spaces and Stein Spaces . . . . . . . . . . . . . . . .403 1. 2. 3. 4. 5. Topological Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 q-Convex Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 q-Convexity Properties in Top Degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 Andreotti-Grauert Finiteness Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 Grauerts Direct Image Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

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Chapter IComplex Dierential Calculus and Pseudoconvexity

This introductive chapter is mainly a review of the basic tools and concepts which will be employed in the rest of the book: dierential forms, currents, holomorphic and plurisubharmonic functions, holomorphic convexity and pseudoconvexity. Our study of holomorphic convexity is principally concentrated here on the case of domains in Cn . The more powerful machinery needed for the study of general complex varieties (sheaves, positive currents, hermitian dierential geometry) will be introduced in Chapters II to V. Although our exposition pretends to be almost self-contained, the reader is assumed to have at least a vague familiarity with a few basic topics, such as dierential calculus, measure theory and distributions, holomorphic functions of one complex variable, . . . . Most of the necessary background can be found in the books of [Rudin 1966] and [Warner 1971]; the basics of distribution theory can be found in Chapter I of [Hrmander 1963]. On the other hand, the reader who has already some knowledge of o complex analysis in several variables should probably bypass this chapter.

1. Dierential Calculus on Manifolds 1.A. Dierentiable Manifolds The notion of manifold is a natural extension of the notion of submanifold dened by a set of equations in Rn . However, as already observed by Riemann during the 19th century, it is important to dene the notion of a manifold in a exible way, without necessarily requiring that the underlying topological space is embedded in an ane space. The precise formal denition was rst introduced by H. Weyl in [Weyl 1913]. Let m N and k N {, }. We denote by k the class of functions which are k-times dierentiable with continuous derivatives if k = , and by C the class of real analytic functions. A dierentiable manifold M of real dimension m and of class k is a topological space (which we shall always assume Hausdor and separable, i.e. possessing a countable basis of the topology), equipped with an atlas of class k with values in Rm . An atlas of class k is a collection of homeomorphisms : U V , I, called dierentiable charts, such that (U )I is an open covering of M and V an open subset of Rm , and such that for all , I the transition map (1.1)1 = : (U U ) (U U )

is a k dieomorphism from an open subset of V onto an open subset of V (see Fig. 1). Then the components (x) = (x , . . . , x ) are called the local coordinates on U dened 1 m by the chart ; they are related by the transition relation x = (x ).

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Chapter I. Complex Dierential Calculus and Pseudoconvexity

V (U U )

UU U

(U U )

U M

V Rm Fig. I-1 Charts and transition maps

If M is open and s N {, }, 0 s k, we denote by C s (, R) the set of 1 functions f of class C s on , i.e. such that f is of class C s on (U ) for each ; if is not open, C s (, R) is the set of functions which have a C s extension to some neighborhood of . A tangent vector at a point a M is by denition a dierential operator acting on functions, of the type C 1 (, R) f f = j1 j m

f (a) xj

in any local coordinate system (x1 , . . . , xm ) on an open set a. We then simply write = j /xj . For every a , the n-tuple (/xj )1 j m is therefore a basis of the tangent space to M at a, which we denote by TM,a . The dierential of a function f at a is the linear form on TM,a dened by dfa () = f = j f /xj (a), TM,a .

In particular dxj () = j and we may consequently write df = (f /xj )dxj . From this, we see that (dx1 , . . . , dxm ) is the dual basis of (/x1 , . . . , /xm ) in the cotangent space TM,a . The disjoint unions TM = xM TM,x and TM = xM TM,x are called the tangent and cotangent bundles of M . If is a vector eld of class C s over , that is, a map x (x) TM,x such that (x) = j (x) /xj has C s coecients, and if is another vector eld of class C s with s 1, the Lie bracket [, ] is the vector eld such that (1.2) [, ] f = ( f ) ( f ). k k j xj xj . xk

In coordinates, it is easy to check that (1.3) [, ] =1 j,k m

j

1. Dierential Calculus on Manifolds

9

1.B. Dierential Forms A dierential form u of degree p, or briey a p-form over M , is a map u on M with values u(x) p TM,x . In a coordinate open set M , a dierential p-form can be written u(x) = uI (x) dxI ,|I|=p

where I = (i1 , . . . , ip ) is a multi-index with integer components, i1 < . . . < ip and dxI := dxi1 . . . dxip . The notation |I| stands for the number of components of I, and is read length of I. For all integers p = 0, 1, . . . , m and s N {}, s k, we denote by C s (M, p TM ) the space of dierential p-forms of class C s , i.e. with C s coecients uI . Several natural operations on dierential forms can be dened. 1.B.1. Wedge Product. If v(x) = vJ (x) dxJ is a q-form, the wedge product of u and v is the form of degree (p + q) dened by (1.4) u v(x) = uI (x)vJ (x) dxI dxJ .

|I|=p,|J|=q

1.B.2. Contraction by a tangent vector. A p-form u can be viewed as an antisymmetric p-linear form on TM . If = j /xj is a tangent vector, we dene the contraction u to be the dierential form of degree p 1 such that (1.5) ( u)(1 , . . . , p1 ) = u(, 1 , . . . , p1 ) u is bilinear and we nd easily if j I, / if j = il I. for all tangent vectors j . Then (, u) xj dxI =

0 (1)l1 dxI

{j}

A simple computation based on the above formula shows that contraction by a tangent vector is a derivation, i.e. (1.6) (u v) = ( u) v + (1)deg u u ( v).

1.B.3. Exterior derivative. This is the dierential operator d : C s (M, p TM ) C s1 (M, p+1 TM )

dened in local coordinates by the formula (1.7) du =|I|=p, 1 k m

uI dxk dxI . xk

Alternatively, one can dene du by its action on arbitrary vector elds 0 , . . . , p on M . The formula is as follows du(0 , . . . , p ) = (1.7 ) +0 j p

(1)j j u(0 , . . . , j , . . . , p ) (1)j+k u([j , k ], 0 , . . . , j , . . . , k , . . . , p ).

0 j }, converges weakly to T as tends to 0. Indeed, T , f = T, f and f converges to f in p () with respect to all seminorms ps . K

20


2.D.4. Poincar Lemma for Currents. Let T s q () be a closed current on an e open set Rm . We rst show that T is cohomologous to a smooth form. In fact, let (Rm ) be a cut-o function such that Supp , 0 < 1 and |d| 1 on . For any vector v B(0, 1) we set Fv (x) = x + (x)v. Since x (x)v is a contraction, Fv is a dieomorphism of Rm which leaves invariant pointwise, so Fv () = . This dieomorphism is homotopic to the identity through the homotopy Hv (t, x) = Ftv (x) : [0, 1] which is proper for every v. Formula (2.22) implies (Fv ) T T = d (Hv ) ([0, 1] T ) . After averaging with a smoothing kernel (v) we get T = dS where =B(0,)

(Fv ) T (v) dv,

S=B(0,)

(Hv ) ([0, 1] T ) (v) dv.p

Then S is a current of the same order s as T and is smooth. Indeed, for u we have , u = T, u where u (x) = Fv u(x) (v) dv ;B(0,)

()

we can make a change of variable z = Fv (x) v = (x)1 (z x) in the last integral and perform derivatives on to see that each seminorm pt (u ) is controlled by the sup K norm of u. Thus and all its derivatives are currents of order 0, so is smooth. Now we have d = 0 and by the usual Poincar lemma (1.22) applied to we obtain e (2.24) Theorem. Let Rm be a starshaped open subset and T s q () a current of degree q 1 and order s such that dT = 0. There exists a current S s q1 () of degree q 1 and order s such that dS = T on .

3. Holomorphic Functions and Complex Manifolds 3.A. Cauchy Formula in One Variable We start by recalling a few elementary facts in one complex variable theory. Let C be an open set and let z = x + iy be the complex variable, where x, y R. If f is a function of class C 1 on , we have df = with the usual notations (3.1) 1 , = i z 2 x y 1 . = +i z 2 x y f f f f dx + dy = dz + dz x y z z

The function f is holomorphic on if df is C-linear, that is, f /z = 0.

3. Holomorphic Functions and Complex Manifolds

21

(3.2) Cauchy formula. Let K C be a compact set with piecewise C 1 boundary K. Then for every f C 1 (K, C) f (w) = 1 2iK

f (z) dz zw

K

1 f d(z), (z w) z

w K

i where d(z) = 2 dz dz = dx dy is the Lebesgue measure on C.

Proof. Assume for simplicity w = 0. As the function z 1/z is locally integrable at z = 0, we get 1 f d(z) = lim 0 z z 1 f i dz dz K D(0,) z z 2 dz 1 f (z) = lim d 0 K D(0,) 2i z dz 1 dz 1 f (z) f (z) lim = 0 2i D(0,) 2i K z z1 2 2 0

K

by Stokes formula. The last integral is equal to as tends to 0.

f (ei ) d and converges to f (0)

When f is holomorphic on , we get the usual Cauchy formula (3.3) f (w) = 1 2i f (z) dz, zw w K ,

K

from which many basic properties of holomorphic functions can be derived: power and Laurent series expansions, Cauchy residue formula, . . . Another interesting consequence is: (3.4) Corollary. The L1 function E(z) = 1/z is a fundamental solution of the loc operator /z on C, i.e. E/z = 0 (Dirac measure at 0). As a consequence, if v is a distribution with compact support in C, then the convolution u = (1/z) v is a solution of the equation u/z = v. Proof. Apply (3.2) with w = 0, f (C) and K Supp f , so that f = 0 on the boundary K and f (0) = 1/z, f /z . (3.5) Remark. It should be observed that this formula cannot be used to solve the equation u/z = v when Supp v is not compact; moreover, if Supp v is compact, a solution u with compact support need not always exist. Indeed, we have a necessary condition v, z n = u, z n /z = 0 for all integers n 0. Conversely, when the necessary condition v, z n = 0 is satised, the canonical solution u = (1/z) v has compact support: this is easily seen by means of the power series expansion (w z)1 = z n wn1 , if we suppose that Supp v is contained in the disk |z| < R and that |w| > R.

22


3.B. Holomorphic Functions of Several Variables Let Cn be an open set. A function f : C is said to be holomorphic if f is continuous and separately holomorphic with respect to each variable, i.e. zj f (. . . , zj , . . .) is holomorphic when z1 , . . . , zj1 , zj+1 , . . . , zn are xed. The set of holomorphic functions on is a ring and will be denoted (). We rst extend the Cauchy formula to the case of polydisks. The open polydisk D(z0 , R) of center (z0,1 , . . . , z0,n ) and (multi)radius R = (R1 , , . . . , Rn ) is dened as the product of the disks of center z0,j and radius Rj > 0 in each factor C : (3.6) D(z0 , R) = D(z0,1 , R1 ) . . . D(z0,n , Rn ) Cn .

The distinguished boundary of D(z0 , R) is by denition the product of the boundary circles (3.7) (z0 , R) = (z0,1 , R1 ) . . . (z0,n , Rn ).

It is important to observe that the distinguished boundary is smaller than the topological boundary D(z0 , R) = j {z D(z0 , R) ; |zj z0,j | = Rj } when n 2. By induction on n, we easily get the (3.8) Cauchy formula on polydisks. If D(z0 , R) is a closed polydisk contained in and f (), then for all w D(z0 , R) we have f (w) = 1 (2i)n(z0 ,R)

f (z1 , . . . , zn ) dz1 . . . dzn . (z1 w1 ) . . . (zn wn )

The expansion (zj wj )1 = (wj z0,j )j (zj z0,j )j 1 , j N, 1 j n, shows that f can be expanded as a convergent power series f (w) = Nn a (w z0 ) over the polydisk D(z0 , R), with the standard notations z = z1 1 . . . zn n , ! = 1 ! . . . n ! and with (3.9) a = 1 (2i)n f (z1 , . . . , zn ) dz1 . . . dzn f () (z0 ) = . (z1 z0,1 )1 +1 . . . (zn z0,n )n +1 !

(z0 ,R)

As a consequence, f is holomorphic over if and only if f is C-analytic. Arguments similar to the one variable case easily yield the (3.10) Analytic continuation theorem. If is connected and if there exists a point z0 such that f () (z0 ) = 0 for all Nn , then f = 0 on . Another consequence of (3.9) is the Cauchy inequality (3.11) |f () (z0 )| ! sup |f |, R (z0 ,R) D(z0 , R) ,

From this, it follows that every bounded holomorphic function on Cn is constant (Liouvilles theorem), and more generally, every holomorphic function F on Cn such that


23

We endow () with the topology of uniform convergence on compact sets K , that is, the topology induced by C 0 (, C). Then () is closed in C 0 (, C). The Cauchy inequalities (3.11) show that all derivations D are continuous operators on () and that any sequence fj () that is uniformly bounded on all compact sets K is locally equicontinuous. By Ascolis theorem, we obtain (3.12) Montels theorem. Every locally uniformly bounded sequence (fj ) in a convergent subsequence (fj() ).

|F (z)| degree

A(1 + |z|)B with suitable constants A, B B.

0 is in fact a polynomial of total

() has

In other words, bounded subsets of the Frchet space () are relatively compact (a e Frchet space possessing this property is called a Montel space). e 3.C. Dierential Calculus on Complex Analytic Manifolds A complex analytic manifold X of dimension dimC X = n is a dierentiable manifold equipped with a holomorphic atlas ( ) with values in Cn ; this means by denition that the transition maps are holomorphic. The tangent spaces TX,x then have a natural complex vector space structure, given by the coordinate isomorphisms d (x) : TX,x Cn , U x ;

the induced complex structure on TX,x is indeed independent of since the dierentials R d are C-linear isomorphisms. We denote by TX the underlying real tangent space R and by End(TX ) the almost complex structure, i.e. the operator of multiplication J by i = 1. If (z1 , . . . , zn ) are complex analytic coordinates on an open subset X R and zk = xk + iyk , then (x1 , y1 , . . . , xn , yn ) dene real coordinates on , and TX admits (/x1 , /y1 , . . ., /xn , /yn) as a basis ; the almost complex structure is given by J(/xk ) = /yk , J(/yk ) = /xk . The complexied tangent space R R R C TX = C R TX = TX iTX splits into conjugate complex subspaces which are the eigenspaces of the complexied endomorphism Id J associated to the eigenvalues i and i. These subspaces have respective bases (3.13) 1 , = i zk 2 xk yk 1 , = +i z k 2 xk yk 1 k n

and are denoted T 1,0 X (holomorphic vectors or vectors of type (1, 0)) and T 0,1 X (antiholomorphic vectors or vectors of type (0, 1)). The subspaces T 1,0 X and T 0,1 X are canonically isomorphic to the complex tangent space TX (with complex structure J) and its conjugate TX (with conjugate complex structure J), via the C-linear embeddings1,0 0,1 TX TX C TX , TX TX C TX 1 1 ( + iJ). 2 ( iJ), 2 0,1 1,0 We thus have a canonical decomposition CTX = TX TX TX TX , and by duality a decomposition R HomR (TX ; C) HomC (C TX ; C) TX TX

24


where TX is the space of C-linear forms and TX the space of conjugate C-linear forms. With these notations, (dxk , dyk ) is a basis of HomR (TR X, C), (dzj ) a basis of TX , (dz j ) a basis of TX , and the dierential of a function f C 1 (, C) can be written n

(3.14)

df =k=1

f f dxk + dyk = xk yk

n

k=1

f f dzk + dz k . zk z k

The function f is holomorphic on if and only if df is C-linear, i.e. if and only if f satises the Cauchy-Riemann equations f /zk = 0 on , 1 k n. We still denote here by (X) the algebra of holomorphic functions on X. Now, we study the basic rules of complex dierential calculus. The complexied R exterior algebra C R (TX ) = (C TX ) is given by R C k (C TX ) = k TX TX

=p+q=k

p,q TX ,

0

k

2n

where the exterior products are taken over C, and where the components p,q TX are dened by

(3.15)

p,q TX = p TX q TX .

A complex dierential form u on X is said to be of bidegree or type (p, q) if its value at every point lies in the component p,q TX ; we shall denote by C s (, p,q TX ) the space of dierential forms of bidegree (p, q) and class C s on any open subset of X. If is a coordinate open set, such a form can be written u(z) =|I|=p,|J|=q

uI,J (z) dzI dz J ,

uI,J C s (, C).

This writing is usually much more convenient than the expression in terms of the real basis (dxI dyJ )|I|+|J|=k which is not compatible with the splitting of k TC X in its (p, q) components. Formula (3.14) shows that the exterior derivative d splits into d = d + d , where d : d : (3.16 ) (3.16 ) d u = d u =I,J 1 k n. (X, p,q TX ) (X, p+1,q TX ), (X, p,q TX ) (X, p,q+1 TX ), uI,J dzk dzI dz J , zk

I,J 1 k n

uI,J dz k dzI dz J . z k

The identity d2 = (d + d )2 = 0 is equivalent to (3.17) d2 = 0, d d + d d = 0, d2 = 0,

since these three operators send (p, q)-forms in (p+2, q), (p+1, q +1) and (p, q +2)-forms, respectively. In particular, the operator d denes for each p = 0, 1, . . . , n a complex, called the Dolbeault complex (X, p,0 TX ) d (X, p,q TX ) d (X, p,q+1 TX )


25

and corresponding Dolbeault cohomology groups (3.18) H p,q (X, C) = Ker d p,q , Im d p,q1

with the convention that the image of d is zero for q = 0. The cohomology group H p,0 (X, C) consists of (p, 0)-forms u = |I|=p uI (z) dzI such that uI /z k = 0 for all I, k, i.e. such that all coecients uI are holomorphic. Such a form is called a holomorphic p-form on X. Let F : X1 X2 be a holomorphic map between complex manifolds. The pullback F u of a (p, q)-form u of bidegree (p, q) on X2 is again homogeneous of bidegree (p, q), because the components Fk of F in any coordinate chart are holomorphic, hence F dzk = dFk is C-linear. In particular, the equality dF u = F du implies (3.19) d F u = F d u, d F u = F d u.

Note that these commutation relations are no longer true for a non holomorphic change of variable. As in the case of the De Rham cohomology groups, we get a pull-back morphism F : H p,q (X2 , C) H p,q (X1 , C). The rules of complex dierential calculus can be easily extended to currents. We use the following notation. (3.20) Denition. There are decompositionsk

(X, C) =p+q=k

p,q

(X, C),

k (X, C)

=p+q=k

p,q (X, C).

The space (X, C) is called the space of currents of bidimension (p, q) and bidegree p,q (n p, n q) on X, and is also denoted np,nq (X, C). 3.D. Newton and Bochner-Martinelli Kernels The Newton kernel is the elementary solution of the usual Laplace operator = 2 /x2 in Rm . We rst recall a construction of the Newton kernel. j Let d = dx1 . . . dxm be the Lebesgue measure on Rm . We denote by B(a, r) the euclidean open ball of center a and radius r in Rm and by S(a, r) = B(a, r) the corresponding sphere. Finally, we set m = Vol B(0, 1) and m1 = mm so that (3.21) Vol B(a, r) = m r m , Area S(a, r) = m1 r m1 .

The second equality follows from the rst by derivation. An explicit computation of 2 the integral Rm e|x| d(x) in polar coordinates shows that m = m/2 /(m/2)! where x! = (x + 1) is the Euler Gamma function. The Newton kernel is then given by: N (x) = 1 log |x| 2 1 N (x) = |x|2m (m 2)m1 if m = 2, if m = 2.

(3.22)

26


The function N (x) is locally integrable on Rm and satises N = 0 . When m = 2, this follows from Cor. 3.4 and the fact that = 4 2 /zz. When m = 2, this can be checked by computing the weak limit0

lim (|x|2 + 2 )1m/2 = lim m(2 m)2 (|x|2 + 2 )1m/20

= m(2 m) Im 0 with Im = Rm (|x|2 + 1)1m/2 d(x). The last equality is easily seen by performing the change of variable y = x in the integral 2 (|x|2 + 2 )1m/2 f (x) d(x) =Rm Rm

(|y|2 + 1)1m/2 f (y) d(y),

where f is an arbitrary test function. Using polar coordinates, we nd that Im = m1 /m and our formula follows. The Bochner-Martinelli kernel is the (n, n 1)-dierential form on Cn with L1 loc coecients dened by (3.23) kBM (z) = cn1 j n

(1)j

z j dz1 . . . dzn dz 1 . . . dz j . . . dz n , |z|2n

cn = (1)n(n1)/2

(n 1)! . (2i)n

(3.24) Lemma. d kBM = 0 on Cn . Proof. Since the Lebesgue measure on Cn is d(z) =1 j n

i i dzj dz j = 2 2

n

(1)

n(n1) 2 dz1

. . . dzn dz 1 . . . dz n ,

we nd d kBM = = (n 1)! n zj d(z) z j |z|2n 1 2 d(z) 2n2 zj z j |z|

1 j n

1 n(n 1)2n

1 j n

= N (z)d(z) = 0 . We let KBM (z, ) be the pull-back of kBM by the map : Cn Cn Cn , (z, ) z . Then Formula (2.19) implies (3.25) d KBM = 0 = [],

where [] denotes the current of integration on the diagonal Cn Cn .


27

(3.26) Koppelman formula. Let Cn be a bounded open set with piecewise C 1 boundary. Then for every (p, q)-form v of class C 1 on we have v(z) = p,q KBM (z, ) v()

+ d z

p,q1 KBM (z, ) v() +

p,q KBM (z, ) d v()

p,q on , where KBM (z, ) denotes the component of KBM (z, ) of type (p, q) in z and (n p, n q 1) in .

Proof. Given w

np,nq

(), we consider the integral KBM (z, ) v() w(z).

It is well dened since KBM has no singularities on Supp v . Since w(z) vanishes on the integral can be extended as well to (). As KBM (z, )v()w(z) is of total bidegree (2n, 2n 1), its dierential d vanishes. Hence Stokes formula yields

KBM (z, ) v() w(z) = =

d KBM (z, ) v() w(z)

p,q d KBM (z, ) v() w(z) KBM (z, ) d v() w(z) p,q1 KBM (z, ) v() d w(z).

(1)p+q By (3.25) we have

d KBM (z, ) v() w(z) =

[] v() w(z) =

v(z) w(z)

Denoting , the pairing between currents and test forms on , the above equality is thus equivalent to KBM (z, ) v(), w(z) = v(z) (1)p+qp,q KBM (z, ) d v(), w(z) p,q1 KBM (z, ) v(), d w(z) ,

which is itself equivalent to the Koppelman formula by integrating d v by parts. (3.27) Corollary. Let v s p,q (Cn ) be a form of class C s with compact support such that d v = 0, q 1. Then the (p, q 1)-form u(z) =Cn p,q1 KBM (z, ) v()

is a C s solution of the equation d u = v. Moreover, if (p, q) = (0, 1) and n 2 then u has 0,1 compact support, thus the Dolbeault cohomology group with compact support Hc (Cn , C) vanishes for n 2.

28


Proof. Apply the Koppelman formula on a suciently large ball = B(0, R) containing Supp v. Then the formula immediately gives d u = v. Observe that the coecients of KBM (z, ) are O(|z |(2n1) ), hence |u(z)| = O(|z|(2n1) ) at innity. If q = 1, then u is holomorphic on Cn B(0, R). Now, this complement is a union of complex lines when n 2, hence u = 0 on Cn B(0, R) by Liouvilles theorem. (3.28) Hartogs extension theorem. Let be an open set in Cn , n 2, and let K be a compact subset such that K is connected. Then every holomorphic function f ( K) extends into a function f (). Proof. Let () be a cut-o function equal to 1 on a neighborhood of K. Set f0 = (1 )f (), dened as 0 on K. Then v = d f0 = f d can be extended by 0 outside , and can thus be seen as a smooth (0, 1)-form with compact support in Cn , such that d v = 0. By Cor. 3.27, there is a smooth function u with compact support in Cn such that d u = v. Then f = f0 u (). Now u is holomorphic outside Supp , so u vanishes on the unbounded component G of Cn Supp . The boundary G is contained in Supp K, so f = (1 )f u coincides with f on the non empty open set G K. Therefore f = f on the connected open set K. A rened version of the Hartogs extension theorem due to Bochner will be given in Exercise 8.13. It shows that f need only be given as a C 1 function on , satisfying the tangential Cauchy-Riemann equations (a so-called CR-function). Then f extends as a holomorphic function f () C 0 (), provided that is connected. 3.E. The Dolbeault-Grothendieck Lemma We are now in a position to prove the Dolbeault-Grothendieck lemma [Dolbeault 1953], which is the analogue for d of the Poincar lemma. The proof given below makes e use of the Bochner-Martinelli kernel. Many other proofs can be given, e.g. by using a reduction to the one dimensional case in combination with the Cauchy formula (3.2), see Exercise 8.5 or [Hrmander 1966]. o (3.29) Dolbeault-Grothendieck lemma. Let be a neighborhood of 0 in Cn and v s p,q (, C), [resp. v s p,q (, C)], such that d v = 0, where 1 s . a) If q = 0, then v(z) = |I|=p vI (z) dzI is a holomorphic p-form, i.e. a form whose coecients are holomorphic functions. b) If q 1, there exists a neighborhood of 0 and a form u in a current u s p,q1 (, C)] such that d u = v on .s p,q1

(, C) [resp.

Proof. We assume that is a ball B(0, r) Cn and take for simplicity r > 1 (possibly after a dilation of coordinates). We then set = B(0, 1). Let () be a cut-o function equal to 1 on . The Koppelman formula (3.26) applied to the form v on gives (z)v(z) = d z p,q1 KBM (z, ) ()v() + p,q KBM (z, ) d () v().

This formula is valid even when v is a current, because we may regularize v as v and take the limit. We introduce on Cn Cn Cn the kernel n (1)j (wj j ) (dzk dk ) (dwk d k ). K(z, w, ) = cn ((z ) (w ))n k j=1 k=j

4. Subharmonic Functions

29

By construction, KBM (z, ) is the result of the substitution w = z in K(z, w, ), i.e. KBM = h K where h(z, ) = (z, z, ). We denote by K p,q the component of K of p,q bidegree (p, 0) in z, (q, 0) in w and (n p, n q 1) in . Then KBM = h K p,q and we nd v = d u0 + g v1 on , where g(z) = (z, z) and u0 (z) = p,q1 KBM (z, ) ()v(),

v1 (z, w) =

K p,q (z, w, ) d () v().

By denition of K p,q (z, w, ), v1 is holomorphic on the open set U = (z, w) ; , Re(z ) (w ) > 0 , / which contains the conjugate-diagonal points (z, z) as well as the points (z, 0) and (0, w) in . Moreover U clearly has convex slices ({z} Cn ) U and (Cn {w}) U . In particular U is starshaped with respect to w, i.e. (z, w) U = (z, tw) U, t [0, 1].

As u1 is of type (p, 0) in z and (q, 0) in w, we get d (g v1 ) = g dw v1 = 0, hence dw v1 = 0. z p,q1 For q = 0 we have KBM = 0, thus u0 = 0, and v1 does not depend on w, thus v is holomorphic on . For q 1, we can use the homotopy formula (1.23) with respect to w (considering z as a parameter) to get a holomorphic form u1 (z, w) of type (p, 0) in z and (q 1, 0) in w, such that dw u1 (z, w) = v1 (z, w). Then we get d g u1 = g dw u1 = g v1 , hence v = d (u0 + g u1 ) on . Finally, the coecients of u0 are obtained as linear combinations of convolutions of the coecients of v with L1 functions of the form j ||2n . Hence u0 is of class C s (resp. loc is a current of order s), if v is. (3.30) Corollary. The operator d is hypoelliptic in bidegree (p, 0), i.e. if a current f p,0 (X, C) satises d f p,1 (X, C), then f p,0 (X, C). Proof. The result is local, so we may assume that X = is a neighborhood of 0 in Cn . The (p, 1)-form v = d f p,1 (X, C) satises d v = 0, hence there exists u p,0 (, C) such that d u = d f . Then f u is holomorphic and f = (f u) + u p,0 (, C).

4. Subharmonic FunctionsA harmonic (resp. subharmonic) function on an open subset of Rm is essentially a function (or distribution) u such that u = 0 (resp. u 0). A fundamental example of subharmonic function is given by the Newton kernel N , which is actually harmonic on Rm {0}. Subharmonic functions are an essential tool of harmonic analysis and potential theory. Before giving their precise denition and properties, we derive a basic integral formula involving the Green kernel of the Laplace operator on the ball.

30


4.A. Construction of the Green Kernel The Green kernel G (x, y) of a smoothly bounded domain Rm is the solution of the following Dirichlet boundary problem for the Laplace operator on : (4.1) Denition. The Green kernel of a smoothly bounded domain Rm is a function G (x, y) : [, 0] with the following properties: a) G (x, y) is

on

Diag

(Diag = diagonal ) ;

b) G (x, y) = G (y, x) ; c) G (x, y) < 0 on and G (x, y) = 0 on ; d) x G (x, y) = y on for every xed y . It can be shown that G always exists and is unique. The uniqueness is an easy consequence of the maximum principle (see Th. 4.14 below). In the case where = B(0, r) is a ball (the only case we are going to deal with), the existence can be shown through explicit calculations. In fact the Green kernel Gr (x, y) of B(0, r) is (4.2) Gr (x, y) = N (x y) N r2 |y| x 2 y r |y| , x, y B(0, r).

A substitution of the explicit value of N (x) yields: |x y|2 1 log 2 if m = 2, otherwise 4 r 2 x, y + r12 |x|2 |y|2 1 1 1m/2 |x y|2m r 2 2 x, y + 2 |x|2 |y|2 . Gr (x, y) = (m 2)m1 r Gr (x, y) = (4.3) Theorem. The above dened function Gr satises all four properties (4.1 ad) on = B(0, r), thus Gr is the Green kernel of B(0, r). Proof. The rst three properties are immediately veried on the formulas, because r 2 2 x, y + 1 2 2 1 |x| |y| = |x y|2 + 2 r 2 |x|2 r 2 |y|2 . 2 r r

For property d), observe that r 2 y/|y|2 B(0, r) whenever y B(0, r) {0}. The second / Newton kernel in the right hand side of (4.1) is thus harmonic in x on B(0, r), and x Gr (x, y) = x N (x y) = y on B(0, r).

4.B. Green-Riesz Representation Formula and Dirichlet Problem 4.B.1. Green-Riesz Formula. For all smooth functions u, v on a smoothly bounded domain Rm , we have (4.4)

(u v v u) d =

u

v u d v


31

where / is the derivative along the outward normal unit vector of and d the euclidean area measure. Indeed (1)j1 dx1 . . . dxj . . . dxm = j d, for the wedge product of , dx with the left hand side is j d. Therefore v d = m j=1

v j d = xj

m

(1)j1j=1

v dx1 . . . dxj . . . dxm . xj

Formula (4.4) is then an easy consequence of Stokes theorem. Observe that (4.4) is still valid if v is a distribution with singular support relatively compact in . For = B(0, r), u C 2 B(0, r), R and v(y) = Gr (x, y), we get the Green-Riesz representation formula: (4.5) u(x) =B(0,r)

u(y) Gr (x, y) d(y) +S(0,r)

u(y) Pr (x, y) d(y)

where Pr (x, y) = Gr (x, y)/(y), (x, y) B(0, r) S(0, r). The function Pr (x, y) is called the Poisson kernel. It is smooth and satises x Pr (x, y) = 0 on B(0, r) by (4.1 d). A simple computation left to the reader yields: (4.6) Pr (x, y) = r 2 |x|2 . m1 r |x y|m 1

Formula (4.5) for u 1 shows that S(0,r) Pr (x, y) d(y) = 1. When x in B(0, r) tends to x0 S(0, r), we see that Pr (x, y) converges uniformly to 0 on every compact subset of S(0, r) {x0 } ; it follows that the measure Pr (x, y) d(y) converges weakly to x0 on S(0, r). 4.B.2. Solution of the Dirichlet Problem. For any bounded measurable function v on S(a, r) we dene (4.7) Pa,r [v](x) =S(a,r)

v(y) Pr (x a, y a) d(y),

x B(a, r).

If u C 0 B(a, r), R C 2 B(a, r), R is harmonic, i.e. u = 0 on B(a, r), then (4.5) gives u = Pa,r [u] on B(a, r), i.e. the Poisson kernel reproduces harmonic functions. Suppose now that v C 0 S(a, r), R is given. Then Pr (x a, y a) d(y) converges weakly to x0 when x tends to x0 S(a, r), so Pa,r [v](x) converges to v(x0 ). It follows that the function u dened by u = Pa,r [v] on B(a, r), u=v on S(a, r) is continuous on B(a, r) and harmonic on B(a, r) ; thus u is the solution of the Dirichlet problem with boundary values v. 4.C. Denition and Basic Properties of Subharmonic Functions

32


4.C.1. Denition. Mean Value Inequalities. If u is a Borel function on B(a, r) which is bounded above or below, we consider the mean values of u over the ball or sphere: (4.8) (4.8 ) B (u ; a, r) = 1 u(x) d(x), m r m B(a,r) 1 u(x) d(x). S (u ; a, r) = m1 r m1 S(a,r)

As d = dr d these mean values are related by (4.9) B (u ; a, r) = 1 m r m1 0 r

m1 tm1 S (u ; a, t) dt0

=m

tm1 S (u ; a, rt) dt.

Now, apply formula (4.5) with x = 0. We get Pr (0, y) = 1/m1 r m1 and Gr (0, y) = r (|y|2m r 2m )/(2 m)m1 = (1/m1 ) |y| t1m dt, thus u(y) Gr (0, y) d(y) = 1 m1r 0 0 r

dt tm1|y| 0 (resp. u(a) < 0), Formula (4.10) shows that S (u ; a, r) > u(a) (resp. S (u ; a, r) < u(a)) for r small enough. In particular, u is harmonic (i.e. u = 0) if and only if u satises the mean value equality S (u ; a, r) = u(a), B(a, r) . Now, observe that if ( ) is a family of radially symmetric smoothing kernels associated with (x) = (|x|) and if u is a Borel locally bounded function, an easy computation yields u (a) =B(0,1) 1

u(a + x) (x) d S (u ; a, t) (t) tm1 dt.0

(4.11)

= m1

Thus, if u is a Borel locally bounded function satisfying the mean value equality on , (4.11) shows that u = u on , in particular u must be smooth. Similarly, if we replace the mean value equality by an inequality, the relevant regularity property to be required for u is just semicontinuity.


33

(4.12) Theorem and denition. Let u : [, +[ be an upper semicontinuous function. The following various forms of mean value inequalities are equivalent: a) u(x) b) u(a) c) u(a) Pa,r [u](x), S (u ; a, r), B (u ; a, r), B(a, r) , B(a, r) ; B(a, r) ; x B(a, r) ;

d) for every a , there exists a sequence (r ) decreasing to 0 such that u(a) B (u ; a, r ) ;

e) for every a , there exists a sequence (r ) decreasing to 0 such that u(a) S (u ; a, r ) .

A function u satisfying one of the above properties is said to be subharmonic on . The set of subharmonic functions will be denoted by Sh(). By (4.10) we see that a function u C 2 (, R) is subharmonic if and only if u 0 : in fact S (u ; a, r) < u(a) for r small if u(a) < 0. It is also clear on the denitions that every (locally) convex function on is subharmonic. Proof. We have obvious implications a) = b) = c) = d) = e), the second and last ones by (4.10) and the fact that B (u ; a, r ) S (u ; a, t) for at least one t ]0, r [. In order to prove e) = a), we rst need a suitable version of the maximum principle. (4.13) Lemma. Let u : [, +[ be an upper semicontinuous function satisfying property 4.12 e). If u attains its supremum at a point x0 , then u is constant on the connected component of x0 in . Proof. We may assume that is connected. Let W = {x ; u(x) < u(x0 )}. W is open by the upper semicontinuity, and distinct from since x0 W . We want to / show that W = . Otherwise W has a non empty connected component W0 , and W0 has a boundary point a . We have a W , thus u(a) = u(x0 ). By assumption 4.12 e), we get u(a) S (u ; a, r ) for some sequence r 0. For r small enough, W0 intersects B(a, r ) and B(a, r ) ; as W0 is connected, we also have S(a, r ) W0 = . Since u u(x0 ) on the sphere S(a, r ) and u < u(x0 ) on its open subset S(a, r ) W0 , we get u(a) S (u ; a, r) < u(x0 ), a contradiction.

34


(4.14) Maximum principle. If u is subharmonic in (in the sense that u satises the weakest property 4.12 e)), then sup u =

lim supz{}

u(z),

and supK u = supK u(z) for every compact subset K . Proof. We have of course lim supz{} u(z) sup u. If the inequality is strict, this means that the supremum is achieved on some compact subset L . Thus, by the upper semicontinuity, there is x0 L such that sup u = supL u = u(x0 ). Lemma 4.13 shows that u is constant on the connected component 0 of x0 in , hence sup u = u(x0 ) =

lim sup0 z0 {}

u(z)

lim supz{}

u(z),

contradiction. The statement involving a compact subset K is obtained by applying the rst statement to = K . Proof of (4.12) e) = a). Let u be an upper semicontinuous function satisfying 4.12 e) and B(a, r) an arbitrary closed ball. One can nd a decreasing sequence of continuous functions vk C 0 S(a, r), R such that lim vk = u. Set hk = Pa,r [vk ] C 0 B(a, r), R . As hk is harmonic on B(a, r), the function u hk satises 4.12 e) on B(a, r). Furthermore lim supxS(a,r) u(x) hk (x) u() vk () 0, so u hk 0 on B(a, r) by Th. 4.14. By monotone convergence, we nd u Pa,r [u] on B(a, r) when k tends to +. 4.C.2. Basic Properties. Here is a short list of the most basic properties. (4.15) Theorem. For any decreasing sequence (uk ) of subharmonic functions, the limit u = lim uk is subharmonic. Proof. A decreasing limit of upper semicontinuous functions is again upper semicontinuous, and the mean value inequalities 4.12 remain valid for u by Lebesgues monotone convergence theorem. (4.16) Theorem. Let u1 , . . . , up Sh() and : Rp R be a convex function such that (t1 , . . . , tp ) is non decreasing in each tj . If is extended by continuity into a function [, +[p [, +[, then (u1 , . . . , up ) Sh(). In particular u1 + + up , max{u1 , . . . , up }, log(eu1 + + eup ) Sh(). Proof. Every convex function is continuous, hence (u1 , . . . , up ) is upper semicontinuous. One can write (t) = sup Ai (t)iI

where Ai (t) = a1 t1 + + ap tp + b is the family of ane functions that dene supporting hyperplanes of the graph of . As (t1 , . . . , tp ) is non-decreasing in each tj , we have aj 0, thus aj uj (x) + b1 j p

B

aj uj + b ; x, r

B (u1 , . . . , up ) ; x, r


35

for every ball B(x, r) . If one takes the supremum of this inequality over all the Ai s , it follows that (u1 , . . . , up ) satises the mean value inequality 4.12 c). In the last example, the function (t1 , . . . , tp ) = log(et1 + + etp ) is convex because 2 j k = e tj tk2 j etj e2

j etj

2

1 j,k p

and

j etj

2

2 j etj e by the Cauchy-Schwarz inequality.

(4.17) Theorem. If is connected and u Sh(), then either u or u L1 (). loc Proof. Note that a subharmonic function is always locally bounded above. Let W be the set of points x such that u is integrable in a neighborhood of x. Then W is open by denition and u > almost everywhere on W . If x W , one can choose a W such that |a x| < r = 1 d(x, ) and u(a) > . Then B(a, r) is a neighborhood of 2 x, B(a, r) and B (u ; a, r) u(a) > . Therefore x W , W is also closed. We must have W = or W = ; in the last case u by the mean value inequality. (4.18) Theorem. Let u Sh() be such that u on each connected component of . Then a) r S (u ; a, r), r B (u ; a, r) are non decreasing functions in the interval ]0, d(a, )[ , and B (u ; a, r) S (u ; a, r). b) For any family ( ) of smoothing kernels, u Sh( ) (u ) is non decreasing in and lim0 u = u.

( , R), the family

Proof. We rst verify statements a) and b) when u C 2 (, R). Then u 0 and S (u ; a, r) is non decreasing in virtue of (4.10). By (4.9), we nd that B (u ; a, r) is also non decreasing and that B (u ; a, r) S (u ; a, r). Furthermore, Formula (4.11) shows that u (a) is non decreasing (provided that is radially symmetric). u u r on r , r > 0,

In the general case, we rst observe that property 4.12 c) is equivalent to the inequality

where r is the probability measure of uniform density on B(0, r). This inequality implies u u r on (r ) = r+ , thus u ( , R) is subharmonic on . It follows that u is non decreasing in ; by symmetry, it is also non decreasing in , and so is u = lim0 u . We have u u by (4.19) and lim sup0 u u by the upper semicontinuity. Hence lim0 u = u. Property a) for u follows now from its validity for u and from the monotone convergence theorem. (4.19) Corollary. If u Sh() is such that u on each connected component of , then u computed in the sense of distribution theory is a positive measure. Indeed (u ) 0 as a function, and (u ) converges weakly to u in (). Corollary 4.19 has a converse, but the correct statement is slightly more involved than for the direct property:

36


(4.20) Theorem. If v () is such that v is a positive measure, there exists a unique function u Sh() locally integrable such that v is the distribution associated to u. We must point out that u need not coincide everywhere with v, even when v is a locally integrable upper semicontinuous function: for example, if v is the characteristic function of a compact subset K of measure 0, the subharmonic representant of v is u = 0. 0, thus v Sh( ). Proof. Set v = v ( , R). Then v = (v) Arguments similar to those in the proof of Th. 4.18 show that (v ) is non decreasing in . Then u := lim0 v Sh() by Th. 4.15. Since v converges weakly to v, the monotone convergence theorem shows that v, f = lim v f d =

0

u f d,

f

(),

f

0,

which concludes the existence part. The uniqueness of u is clear from the fact that u must satisfy u = lim u = lim v . The most natural topology on the space Sh() of subharmonic functions is the topology induced by the vector space topology of L1 () (Frchet topology of convergence e loc 1 in L norm on every compact subset of ). (4.21) Proposition. The convex cone Sh() L1 () is closed in L1 (), and it has loc loc the property that every bounded subset is relatively compact. Proof. Let (uj ) be a sequence in Sh() L1 (). If uj u in L1 () then uj u loc loc in the weak topology of distributions, hence u 0 and u can be represented by a subharmonic function thanks to Th. 4.20. Now, suppose that uj L1 (K) is uniformly bounded for every compact subset K of . Let j = uj 0. If () is a test function equal to 1 on a neighborhood of K and such that 0 1 on , we nd j (K)

uj d =

uj d

C uj

L1 (K ) ,

where K = Supp , hence the sequence of measures (j ) is uniformly bounded in mass on every compact subset of . By weak compactness, there is a subsequence (j ) which converges weakly to a positive measure on . We claim that f (j ) converges to f () in L1 (Rm ) for every function f L1 (Rm ). In fact, this is clear if f (Rm ), loc loc and in general we use an approximation of f by a smooth function g together with the estimate (f g) (j )L1 (A)

(f g)

L1 (A+K ) j (K

),

A Rm

to get the conclusion. We apply this when f = N is the Newton kernel. Then hj = uj N (j ) is harmonic on and bounded in L1 (). As hj = hj for any smoothing kernel , we see that all derivatives D hj = hj (D ) are in fact uniformly locally bounded in . Hence, after extracting a new subsequence, we may suppose that


37

hj converges uniformly to a limit h on . Then uj = hj + N (j ) converges to u = h + N () in L1 (), as desired. loc We conclude this subsection by stating a generalized version of the Green-Riesz formula. (4.22) Proposition. Let u Sh() L1 () and B(0, r) . loc a) The Green-Riesz formula still holds true for such an u, namely, for every x B(0, r) u(x) =B(0,r)

u(y) Gr (x, y) d(y) +S(0,r)

u(y) Pr (x, y) d(y).

b) (Harnack inequality) If u 0 on B(0, r), then for all x B(0, r) 0 If u u(x)S(0,r)

u(y) Pr (x, y) d(y)

r m2 (r + |x|) S (u ; 0, r). (r |x|)m1

0 on B(0, r), then for all x B(0, r) u(x)S(0,r)

u(y) Pr (x, y) d(y)

r m2 (r |x|) S (u ; 0, r) (r + |x|)m1

0.

Proof. We know that a) holds true if u is of class C 2 . In general, we replace u by u and take the limit. We only have to check that (y) Gr (x, y) d(y) = limB(0,r) 0

(y) Gr (x, y) d(y)B(0,r)

for the positive measure = u. Let us denote by Gx (y) the function such that Gx (y) = Then (y) Gr (x, y) d(y) =B(0,r) Rm

Gr (x, y) if x B(0, r) 0 if x B(0, r). / (y) Gx (y) d(y) (y) Gx (y) d(y).Rm

=

However Gx is continuous on Rm {x} and subharmonic in a neighborhood of x, hence Gx converges uniformly to Gx on every compact subset of Rm {x}, and converges pointwise monotonically in a neighborhood of x. The desired equality follows by the monotone convergence theorem. Finally, b) is a consequence of a), for the integral involving u is nonpositive and 1 r m2 (r |x|) m1 r m1 (r + |x|)m1 Pr (x, y) 1 r m2 (r + |x|) m1 r m1 (r |x|)m1 |x y|m (r + |x|)m .

by (4.6) combined with the obvious inequality (r |x|)m

38


4.C.3. Upper Envelopes and Choquets Lemma. Let Rn and let (u )I be a family of upper semicontinuous functions [, +[. We assume that (u ) is locally uniformly bounded above. Then the upper envelope u = sup u need not be upper semicontinuous, so we consider its upper semicontinuous regularization: u (z) = lim sup u u(z).0 B(z,)

It is easy to check that u is the smallest upper semicontinuous function which is u. Our goal is to show that u can be computed with a countable subfamily of (u ). Let B(zj , j ) be a countable basis of the topology of . For each j, let (zjk ) be a sequence of points in B(zj , j ) such that sup u(zjk ) =k

sup u,B(zj ,j )

and for each pair (j, k), let (j, k, l) be a sequence of indices I such that u(zjk ) = supl u(j,k,l) (zjk ). Set v = sup u(j,k,l) .j,k,l

Then v

u and v sup vB(zj ,j )

u . On the other hand sup v(zjk )k

sup u(j,k,l) (zjk ) = sup u(zjk ) =k,l k

sup u.B(zj ,j )

As every ball B(z, ) is a union of balls B(zj , j ), we easily conclude that v v = u . Therefore:

u , hence

(4.23) Choquets lemma. Every family (u ) has a countable subfamily (vj ) = (u(j) ) such that its upper envelope v satises v u u = v . (4.24) Proposition. If all u are subharmonic, the upper regularization u is subharmonic and equal almost everywhere to u. Proof. By Choquets lemma we may assume that (u ) is countable. Then u = sup u is a Borel function. As each u satises the mean value inequality on every ball B(z, r) , we get u(z) = sup u (z) sup B (u ; z, r) B (u ; z, r). The right-hand side is a continuous function of z, so we infer u (z) B (u ; z, r) B (u ; z, r)

and u is subharmonic. By the upper semicontinuity of u and the above inequality we nd u (z) = limr0 B (u ; z, r), thus u = u almost everywhere by Lebesgues lemma.

5. Plurisubharmonic Functions

39

5. Plurisubharmonic Functions 5.A. Denition and Basic Properties Plurisubharmonic functions have been introduced independently by [Lelong 1942] and [Oka 1942] for the study of holomorphic convexity. They are the complex counterparts of subharmonic functions. (5.1) Denition. A function u : [, +[ dened on an open subset Cn is said to be plurisubharmonic if a) u is upper semicontinuous ; b) for every complex line L Cn , uL is subharmonic on L. The set of plurisubharmonic functions on is denoted by Psh(). An equivalent way of stating property b) is: for all a , Cn , || < d(a, ), then (5.2) u(a) 1 22

u(a + ei ) d.0

An integration of (5.2) over S(0, r) yields u(a) (5.3) Psh() Sh().

S (u ; a, r), therefore

The following results have already been proved for subharmonic functions and are easy to extend to the case of plurisubharmonic functions: (5.4) Theorem. For any decreasing sequence of plurisubharmonic functions uk Psh(), the limit u = lim uk is plurisubharmonic on . (5.5) Theorem. Let u Psh() be such that u on every connected component of . If ( ) is a family of smoothing kernels, then u is and plurisubharmonic on , the family (u ) is non decreasing in and lim0 u = u. (5.6) Theorem. Let u1 , . . . , up Psh() and : Rp R be a convex function such that (t1 , . . . , tp ) is non decreasing in each tj . Then (u1 , . . . , up ) is plurisubharmonic on . In particular u1 + +up , max{u1 , . . . , up }, log(eu1 + +eup ) are plurisubharmonic on . (5.7) Theorem. Let {u } Psh() be locally uniformly bounded from above and u = sup u . Then the regularized upper envelope u is plurisubharmonic and is equal to u almost everywhere. Proof. By Choquets lemma, we may assume that (u ) is countable. Then u is a Borel function which clearly satises (5.2), and thus u also satises (5.2). Hence u is plurisubharmonic. By Proposition 4.24, u = u almost everywhere and u is subharmonic, so u = lim u = lim u

40


is plurisubharmonic. If u C 2 (, R), the subharmonicity of restrictions of u to complex lines, C w u(a + w), a , Cn , is equivalent to 2 u(a + w) = ww 2u (a + w) j k zj z k 0.

1 j,k n

Therefore, u is plurisubharmonic on if and only if 2 u/zj z k (a) j k is a semipositive hermitian form at every point a . This equivalence is still true for arbitrary plurisubharmonic functions, under the following form: (5.8) Theorem. If u Psh(), u on every connected component of , then for all Cn 2u Hu() := j k () zj z k1 j,k n

is a positive measure. Conversely, if v () is such that Hv() is a positive measure for every Cn , there exists a unique function u Psh() locally integrable on such that v is the distribution associated to u. Proof. If u Psh(), then Hu() = weak lim H(u )() 0. Conversely, Hv 0 implies H(v ) = (Hv) 0, thus v Psh(), and also v 0, hence (v ) is non decreasing in and u = lim0 v Psh() by Th. 5.4. (5.9) Proposition. The convex cone Psh() L1 () is closed in L1 (), and it has loc loc the property that every bounded subset is relatively compact. 5.B. Relations with Holomorphic Functions In order to get a better geometric insight, we assume more generally that u is a C 2 function on a complex n-dimensional manifold X. The complex Hessian of u at a point a X is the hermitian form on TX dened by (5.10) Hua =1 j,k n

2u (a) dzj dz k . zj z k

If F : X Y is a holomorphic mapping and if v C 2 (Y, R), we have d d (v F ) = F d d v. In equivalent notations, a direct calculation gives for all TX,a H(v F )a () = 2 v F (a) Fl a) Fm a) j k = HvF (a) F (a). . zl z m zj zk

j,k,l,m

In particular Hua does not depend on the choice of coordinates (z1 , . . . , zn ) on X, and Hva 0 on Y implies H(v F )a 0 on X. Therefore, the notion of plurisubharmonic function makes sense on any complex manifold. (5.11) Theorem. If F : X Y is a holomorphic map and v Psh(Y ), then v F Psh(X).


41

Proof. It is enough to prove the result when X = 1 Cn and X = 2 Cp are open subsets . The conclusion is already known when v is of class C 2 , and it can be extended to an arbitrary upper semicontinuous function v by using Th. 5.4 and the fact that v = lim v . (5.12) Example. By (3.22) we see that log |z| is subharmonic on C, thus log |f | Psh(X) for every holomorphic function f (X). More generally for every fj (X) and j log |f1 |1 + + |fq |q Psh(X) 0 (apply Th. 5.6 with uj = j log |fj | ).

5.C. Convexity Properties The close analogy of plurisubharmonicity with the concept of convexity strongly suggests that there are deeper connections between these notions. We describe here a few elementary facts illustrating this philosophy. Another interesting connection between plurisubharmonicity and convexity will be seen in 7.B (Kiselmans minimum principle). (5.13) Theorem. If = + i where , are open subsets of Rn , and if u(z) is a plurisubharmonic function on that depends only on x = Re z, then x u(x) is convex.1 Proof. This is clear when u C 2 (, R), for 2 u/zj z k = 4 2 u/xj xk . In the general case, write u = lim u and observe that u (z) depends only on x.

(5.14) Corollary. If u is a plurisubharmonic function in the open polydisk D(a, R) = D(aj , Rj ) Cn , then2 1 u(a1 + r1 ei1 , . . . , an + rn ein ) d1 . . . dn , (2)n 0 m(u ; r1 , . . . , rn ) = sup u(z1 , . . . , zn ), rj < R j

(u ; r1 , . . . , rn ) =

zD(a,r)

are convex functions of (log r1 , . . . , log rn ) that are non decreasing in each variable. Proof. That is non decreasing follows from the subharmonicity of u along every coordinate axis. Now, it is easy to verify that the functions2 1 u(a1 + ez1 ei1 , . . . , an + ezn ein ) d1 . . . dn , n (2) 0 m(z1 , . . . , zn ) = sup u(a1 + ez1 w1 , . . . , an + ezn wn )

(z1 , . . . , zn ) =

|wj | 1

are upper semicontinuous, satisfy the mean value inequality, and depend only on Re zj ]0, log Rj [. Therefore and M are convex. Cor. 5.14 follows from the equalities (u ; r1 , . . . , rn ) = (log r1 , . . . , log rn ), m(u ; r1 , . . . , rn ) = m(log r1 , . . . , log rn ).

42


5.D. Pluriharmonic Functions Pluriharmonic functions are the counterpart of harmonic functions in the case of functions of complex variables: (5.15) Denition. A function u is said to be pluriharmonic if u and u are plurisubharmonic. A pluriharmonic function is harmonic (in particular smooth) in any C-analytic coordinate system, and is characterized by the condition Hu = 0, i.e. d d u = 0 or 2 u/zj z k = 0 for all j, k.

If f (X), it follows that the functions Re f, Im f are pluriharmonic. Conversely:1 (5.16) Theorem. If the De Rham cohomology group HDR (X, R) is zero, every pluriharmonic function u on X can be written u = Re f where f is a holomorphic function on X. 1 Proof. By hypothesis HDR (X, R) = 0, u (X) and d(d u) = d d u = 0, hence there (X) such that dg = d u. Then dg is of type (1, 0), i.e. g (X) and exists g

d(u 2 Re g) = d(u g g) = (d u dg) + (d u dg) = 0. Therefore u = Re(2g + C), where C is a locally constant function. 5.E. Global Regularization of Plurisubharmonic Functions We now study a very ecient regularization and patching procedure for continuous plurisubharmonic functions, essentially due to [Richberg 1968]. The main idea is contained in the following lemma: (5.17) Lemma. Let u Psh( ) where X is a locally nite open covering of X. Assume that for every index lim sup u () < max {u (z)}z z

at all points z . Then the function u(z) = max u (z) z

is plurisubharmonic on X. Proof. Fix z0 X. Then the indices such that z0 or z0 do not contribute / to the maximum in a neighborhood of z0 . Hence there is a a nite set I of indices such that z0 and a neighborhood V I on which u(z) = maxI u (z). Therefore u is plurisubharmonic on V . The above patching procedure produces functions which are in general only continuous. When smooth functions are needed, one has to use a regularized max function. Let


43

and

(R, R) be a nonnegative function with support in [1, 1] such that h(h) dh = 0. R

R

(h) dh = 1

(5.18) Lemma. For arbitrary = (1 , . . . , p ) ]0, +[p, the function M (t1 , . . . , tp ) =Rn

max{t1 + h1 , . . . , tp + hp }

(hj /j ) dh1 . . . dhp1 j n

possesses the following properties: a) M (t1 , . . . , tp ) is non decreasing in all variables, smooth and convex on Rn ; b) max{t1 , . . . , tp } M (t1 , . . . , tp ) max{t1 + 1 , . . . , tp + p } ;

c) M (t1 , . . . , tp ) = M(1 ,...,j ,...,p ) (t1 , . . . , tj , , . . . , tp ) if tj + j maxk=j {tk k } ; d) M (t1 + a, . . . , tp + a) = M (t1 , . . . , tp ) + a, a R ; e) if u1 , . . . , up are plurisubharmonic and satisfy H(uj )z () z () where z z is a continuous hermitian form on TX , then u = M (u1 , . . . , up ) is plurisubharmonic and satises Huz () z (). Proof. The change of variables hj hj tj shows that M is smooth. All properties are immediate consequences of the denition, except perhaps e). That M (u1 , . . . , up ) is plurisubharmonic follows from a) and Th. 5.6. Fix a point z0 and > 0. All functions u (z) = uj (z) z0 (z z0 ) + |z z0 |2 are plurisubharmonic near z0 . It follows that j M (u , . . . , u ) = u z0 (z z0 ) + |z z0 |2 1 p is also plurisubharmonic near z0 . Since > 0 was arbitrary, e) follows. (5.19) Corollary. Let u ( ) Psh( ) where X is a locally nite open covering of X. Assume that u (z) < max{u (z)} at every point z , when runs over the indices such that z. Choose a family ( ) of positive numbers so small that u (z) + max z {u (z) } for all and z . Then the function dened by u(z) = M( ) u (z) is smooth and plurisubharmonic on X. (5.20) Denition. A function u Psh(X) is said to be strictly plurisubharmonic if u L1 (X) and if for every point x0 X there exists a neighborhood of x0 and c > 0 loc such that u(z) c|z|2 is plurisubharmonic on , i.e. ( 2 u/zj z k )j k c||2 (as n distributions on ) for all C . (5.21) Theorem ([Richberg 1968]). Let u Psh(X) be a continuous function which is strictly plurisubharmonic on an open subset X, with Hu for some continuous positive hermitian form on . For any continuous function C 0 (), > 0, there exists a plurisubharmonic function u in C 0 (X) () such that u u u + on for such that z

44


and u = u on X , which is strictly plurisubharmonic on and satises H u (1). In particular, u can be chosen strictly plurisubharmonic on X if u has the same property. Proof. Let ( ) be a locally nite open covering of by relatively compact open balls contained in coordinate patches of X. Choose concentric balls of respec tive radii r < r < r and center z = 0 in the given coordinates z = (z1 , . . . , zn ) near , such that still cover . We set 2 u (z) = u (z) + (r |z|2 )

on . (1 )

For < ,0 and < ,0 small enough, we have u u + /2 and Hu on . Set 2 2 2 2 = min{r r , (r r )/2}.

Choose rst < ,0 such that < min /2, and then < ,0 so small that u u < u + on . As (r 2 |z|2 ) is 2 on and > on , we have u < u on and u > u + on , so that the condition required in Corollary 5.19 is satised. We dene u= u M( ) (u ) on X on . ,

By construction, u is smooth on and satises u u u + , Hu (1 ) thanks to 5.18 (b,e). In order to see that u is plurisubharmonic on X, observe that u is the uniform limit of u with u = max u , M(1 ... ) (u1 . . . u ) and u = u on the complement. 5.F. Polar and Pluripolar Sets. Polar and pluripolar sets are sets of poles of subharmonic and plurisubharmonic functions. Although these functions possess a large amount of exibility, pluripolar sets have some properties which remind their loose relationship with holomorphic functions. (5.22) Denition. A set A Rm (resp. A X, dimC X = n) is said to be polar (resp. pluripolar) if for every point x there exist a connected neighborhood W of x and u Sh(W ) (resp. u Psh(W )), u , such that A W {x W ; u(x) = }. Theorem 4.17 implies that a polar or pluripolar set is of zero Lebesgue measure. Now, we prove a simple extension theorem. (5.23) Theorem. Let A be a closed polar set and v Sh( A) such that v is bounded above in a neighborhood of every point of A. Then v has a unique extension v Sh(). Proof. The uniqueness is clear because A has zero Lebesgue measure. On the other hand, every point of A has a neighborhood W such that A W {x W ; u(x) = }, u Sh(W ), u . on1

6. Domains of Holomorphy and Stein Manifolds

45

After shrinking W and subtracting a constant to u, we may assume u 0. Then for every > 0 the function v = v + u Sh(W A) can be extended as an upper semicontinuous on W by setting v = on A W . Moreover, v satises the mean value inequality v (a) S (v ; a, r) if a W A, r < d(a, A W ), and also clearly if a A, r < d(a, W ). Therefore v Sh(W ) and v = (sup v ) Sh(W ). Clearly v coincides with v on W A. A similar proof gives: (5.24) Theorem. Let A be a closed pluripolar set in a complex analytic manifold X. Then every function v Psh(X A) that is locally bounded above near A extends uniquely into a function v Psh(X). (5.25) Corollary. Let A X be a closed pluripolar set. Every holomorphic function f (X A) that is locally bounded near A extends to a holomorphic function f (X). Proof. Apply Th. 5.24 to Re f and Im f . It follows that Re f and Im f have pluriharmonic extensions to X, in particular f extends to f (X). By density of X A, d f = 0 on X. (5.26) Corollary. Let A (resp. A X) be a closed (pluri)polar set. If (resp. X) is connected, then A (resp. X A) is connected. Proof. If A (resp. X A) is a disjoint union 1 2 of non empty open subsets, the function dened by f 0 on 1 , f 1 on 2 would have a harmonic (resp. holomorphic) extension through A, a contradiction.

6. Domains of Holomorphy and Stein Manifolds 6.A. Domains of Holomorphy in Cn . Examples Loosely speaking, a domain of holomorphy is an open subset in Cn such that there is no part of across which all functions f () can be extended. More precisely: (6.1) Denition. Let Cn be an open subset. is said to be a domain of holomorphy if for every connected open set U Cn which meets and every connected component V of U there exists f () such that fV has no holomorphic extension to U . Under the hypotheses made on U , we have = V U . In order to show that is a domain of holomorphy, it is thus sucient to nd for every z0 a function f () which is unbounded near z0 . (6.2) Examples. Every open subset C is a domain of holomorphy (for any z0 , f (z) = (zz0 )1 cannot be extended at z0 ). In Cn , every convex open subset is a domain of holomorphy: if Re z z0 , 0 = 0 is a supporting hyperplane of at z0 , the function f (z) = ( z z0 , 0 )1 is holomorphic on but cannot be extended at z0 . (6.3) Hartogs gure. Assume that n 2. Let Cn1 be a connected open set and an open subset. Consider the open sets in Cn : = (D(R) D(r)) D(R) = D(R) (Hartogs gure), (lled Hartogs gure).

46


where 0

r < R and D(r) C denotes the open disk of center 0 and radius r in C. Cn1 z (1 , z ) 0 R z1 r C z

Fig. I-3 Hartogs gure Then every function f () can be extended to = D(R) by means of the Cauchy formula: f (z1 , z ) = 1 2i f (1 , z ) d1 , 1 z1 z , max{|z1 |, r} < < R.

|1 |=

In fact f (D(R) ) and f = f on D(R) , so we must have f = f on since is connected. It follows that is not a domain of holomorphy. Let us quote two interesting consequences of this example. (6.4) Corollary (Riemanns extension theorem). Let X be a complex analytic manifold, and S a closed submanifold of codimension 2. Then every f (X S) extends holomorphically to X. Proof. This is a local result. We may choose coordinates (z1 , . . . , zn ) and a polydisk D(R)n in the corresponding chart such that S D(R)n is given by equations z1 = . . . = zp = 0, p = codim S 2. Then, denoting = D(R)n1 and = {z2 = . . . = zp = 0}, the complement D(R)n S can be written as the Hartogs gure D(R)n S = (D(R) {0}) D(R) .

It follows that f can be extended to = D(R)n . 6.B. Holomorphic Convexity and Pseudoconvexity Let X be a complex manifold. We rst introduce the notion of holomorphic hull of a compact set K X. This can be seen somehow as the complex analogue of the notion of (ane) convex hull for a compact set in a real vector space. It is shown that domains of holomorphy in Cn are characterized a property of holomorphic convexity. Finally, we prove that holomorphic convexity implies pseudoconvexity a complex analogue of the geometric notion of convexity.


47

(6.5) Denition. Let X be a complex manifold and let K be a compact subset of X. Then the holomorphic hull of K in X is dened to be K = K(X) = z X ; |f (z)| (6.6) Elementary properties. a) K is a closed subset of X containing K. Moreover we have sup |f | = sup |f |,K K

sup |f |, f (X) .K

f (X),

hence K = K. b) If h : X Y is a holomorphic map and K X is a compact set, then h(K(X) ) h(K)(Y ) . In particular, if X Y , then K(X) K(Y ) X. This is immediate from the denition.

c) K contains the union of K with all relatively compact connected components of X K (thus K lls the holes of K). In fact, for every connected component U of X K we have U K, hence if U is compact the maximum principle yields sup |f | = sup |f |U U

sup |f |,K

for all f (X).

d) More generally, suppose that there is a holomorphic map h : U X dened on a relatively compact open set U in a complex manifold S, such that h extends as a continuous map h : U X and h(U ) K. Then h(U ) K. Indeed, for f (X), the maximum principle again yields sup |f h| = sup |f h|U U

sup |f |.K

This is especially useful when U is the unit disk in C. e) Suppose that X = Cn is an open set. By taking f (z) = exp(A(z)) where A is an arbitrary ane function, we see that K() is contained in the intersection of all ane half-spaces containing K. Hence K() is contained in the ane convex hull Ka . As a consequence K() is always bounded and K(Cn ) is a compact set. However, when is arbitrary, K() is not always compact; for example, in case = Cn {0}, n 2, then () = (Cn ) and the holomorphic hull of K = S(0, 1) is the non compact set K = B(0, 1) {0}. (6.7) Denition. A complex manifold X is said to be holomorphically convex if the holomorphic hull K(X) of every compact set K X is compact. (6.8) Remark. A complex manifold X is holomorphically convex if and only if there is an exhausting sequence of holomorphically compact subsets K X, i.e. compact sets such that X= K , K = K , K K1 .

48


Indeed, if X is holomorphically convex, we may dene K inductively by K0 = and K+1 = (K L )(X) , where K is a neighborhood of K and L a sequence of compact sets of X such that X = L . The converse is obvious: if such a sequence (K ) exists, then every compact subset K X is contained in some K , hence K K = K is compact. We now concentrate on domains of holomorphy in Cn . We denote by d and B(z, r) the distance and the open balls associated to an arbitrary norm on Cn , and we set for simplicity B = B(0, 1). (6.9) Proposition. If is a domain of holomorphy and K is a compact subset, then d(K, ) = d(K, ) and K is compact. Proof. Let f (). Given r < d(K, ), we denote by M the supremum of |f | on the compact subset K + rB . Then for every z K and B, the function+

(6.10)

C t f (z + t) =

k=0

1 k D f (z)()k tk k!

is analytic in the disk |t| < r and bounded by M . The Cauchy inequalities imply |Dk f (z)()k | M k! r k , z K, B.

As the left hand side is an analytic fuction of z in , the inequality must also hold for z K, B. Every f () can thus be extended to any ball B(z, r), z K, by means of the power series (6.10). Hence B(z, r) must be contained in , and this shows that d(K, ) r. As r < d(K, ) was arbitrary, we get d(K, ) d(K, ) and the converse inequality is clear, so d(K, ) = d(K, ). As K is bounded and closed in , this shows that K is compact. (6.11) Theorem. Let be an open subset of Cn . The following properties are equivalent: a) is a domain of holomorphy; b) is holomorphically convex; c) For every countable subset {zj }jN without accumulation points in and every sequence of complex numbers (aj ), there exists an interpolation function F () such that F (zj ) = aj . d) There exists a function F point of .

() which is unbounded on any neighborhood of any

Proof. d) = a) is obvious and a) = b) is a consequence of Prop. 6.9. c) = d). If = Cn there is nothing to prove. Otherwise, select a dense sequence (j ) in and take zj such that d(zj , j ) < 2j . Then the interpolation function F () such that F (zj ) = j satises d).


49

b) = c). Let K be an exhausting sequence of holomorphically convex compact sets as in Remark 6.8. Let (j) be the unique index such that zj K(j)+1 K(j) . By the denition of a holomorphic hull, we can nd a function gj () such thatK(j)

sup |gj | < |gj (zj )|.

After multiplying gj by a constant, we may assume that gj (zj ) = 1. Let Pj C[z1 , . . . , zn ] be a polynomial equal to 1 at zj and to 0 at z0 , z1 , . . . , zj1 . We set+

F =j=0

j Pj gj j ,

m

where j C and mj N are chosen inductively such that j = aj m |j Pj gj j |

k Pk (zj )gk (zj )mk ,0 k 1. Then |g,a (z)| > 1 in a neighborhood of a ; by the Borel-Lebesgue lemma, one can nd nitely many functions (g,a )aI such that max |g,a (z)| > 1 for z L ,aI

max |g,a (z)| < 1 for z K .aI

For a suciently large exponent p() we get |g,a |2p() on L ,aI

aI

|g,a |2p()

2 on K .

It follows that the series (z) =N aI

|g,a (z)|2p()

converges uniformly to a real analytic function Psh(X) (see Exercise 8.11). By construction (z) for z L , hence is an exhaustion. (6.15) Example. The converse to Theorem 6.14 does not hold. In fact let X = C2 / be the quotient of C2 by the free abelian group of rank 2 generated by the ane automorphisms g1 (z, w) = (z + 1, ei1 w), g2 (z, w) = (z + i, ei2 w), 1 , 2 R.

Since acts properly discontinuously on C2 , the quotient has a structure of a complex (non compact) 2-dimensional manifold. The function w |w|2 is -invariant, hence it induces a function ((z, w) ) = |w|2 on X which is in fact a plurisubharmonic exhaustion function. Therefore X is weakly pseudoconvex. On the other hand, any holomorphic function f (X) corresponds to a -invariant holomorphic function f (z, w) on C2 . Then z f (z, w) is bounded for w xed, because f (z, w) lies in the image of the compact set K D(0, |w|), K = unit square in C. By Liouvilles theorem, f (z, w) does not depend on z. Hence functions f (X) are in one-to-one correspondence with holomorphic functions f (w) on C such that f (eij w) = f (w). By looking at the Taylor expansion at the origin, we conclude that f must be a constant if 1 Q or 1 Q (if / / 1 , 2 Q and m is the least common denominator of 1 , 2 , then f is a power series of the form k wmk ). From this, it follows easily that X is holomorphically convex if and only if 1 , 2 Q. 6.C. Stein Manifolds The class of holomorphically convex manifolds contains two types of manifolds of a rather dierent nature: domains of holomorphy X = Cn ; In the rst case we have a lot of holomorphic functions, in fact the functions in () separate any pair of points of . On the other hand, if X is compact and connected, the sets Psh(X) and (X) consist of constant functions merely (by the maximum principle). It is therefore desirable to introduce a clear distinction between these two subclasses. For compact complex manifolds.


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this purpose, [Stein 1951] introduced the class of manifolds which are now called Stein manifolds. (6.16) Denition. A complex manifold X is said to be a Stein manifold if a) X is holomorphically convex ; b)

(X) locally separates points in X, i.e. every point x X has a neighborhood V that for any y V {x} there exists f (X) with f (y) = f (x).

such

The second condition is automatic if X = is an open subset of Cn . Hence an open set Cn is Stein if and only if is a domain of holomorphy. (6.17) Lemma. If a complex manifold X satises the axiom (6.16 b) of local separation, there exists a smooth nonnegative strictly plurisubharmonic function u Psh(X). Proof. Fix x0 X. We rst show that there exists a smooth nonnegative function u0 Psh(X) which is strictly plurisubharmonic on a neighborhood of x0 . Let (z1 , . . . , zn ) be local analytic coordinates centered at x0 , and if necessary, replace zj by zj so that the closed unit ball B = { |zj |2 1} is contained in the neighborhood V x0 on which (6.16 b) holds. Then, for every point y B, there exists a holomorphic function f (X) such that f (y) = f (x0). Replacing f with (f f (x0)), we can achieve f (x0) = 0 and |f (y)| > 1. By compactness of B, we nd nitely many functions f1 , . . . , fN (X) such that v0 = |fj |2 satises v0 (x0 ) = 0, while v0 1 on B. Now, we set u0 (z) = v0 (z) on X 2 M {v0 (z), (|z| + 1)/3} on B. B,

where M are the regularized max functions dened in 5.18. Then u0 is smooth and plurisubharmonic, coincides with v0 near B and with (|z|2 + 1)/3 on a neighborhood of x0 . We can cove

complex geometry - jean-pierre demailly

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