Analytic and Algebraic Geometry: Common Problems, Different MethodsJeffery D. McNeal, Mircea Mustata "Analytic and algebraic geometers often study the same geometric structures but bring different methods to bear on them. While this dual approach has been spectacularly successful at solving problems, the language differences between algebra and analysis also represent a difficulty for students and researchers in geometry, particularly complex geometry. The PCMI program was designed to partially address this language gulf, by presenting some of the active developments in algebraic and analytic geometry in a form suitable for students on the 'other side' of the analysis-algebra language divide. One focal point of the summer school was multiplier ideals, a subject of wide current interest in both subjects. The present volume is based on a series of lectures at the PCMI summer school on analytic and algebraic geometry. The series is designed to give a high-level introduction to the advanced techniques behind some recent developments in algebraic and analytic geometry. The lectures contain many illustrative examples, detailed computations, and new perspectives on the topics presented, in order to enhance access of this material to non-specialists."--Publisher's description. |
Contents
3 | |
19 | |
The Bergman kernel | 43 |
Lecture 5 | 51 |
Adjunction and extension from divisors | 59 |
Deformational invariance of plurigenera | 71 |
John P DAngelo Real and Complex Geometry meet the CauchyRiemann Equations 77 | 77 |
Preface | 79 |
Squared norms and proper mappings between balls | 171 |
Holomorphic line bundles | 173 |
Some open problems | 175 |
Basic notions in complex geometry | 189 |
The Hörmander theorem | 217 |
The L2 extension theorem | 251 |
The Skoda division theorem | 277 |
Bibliography | 293 |
Background material 81 81 82 | 81 |
Differential forms | 82 |
Solving the CauchyRiemann equations | 84 |
Complex varieties in real hypersurfaces | 87 |
Hermitian symmetry and polarization | 89 |
Holomorphic decomposition | 90 |
Real analytic hypersurfaces and subvarieties | 98 |
Complex varieties local algebra and multiplicities | 99 |
Pseudoconvexity the Levi form and points of finite type | 105 |
The Levi form | 107 |
Higher order commutators | 111 |
Points of finite type | 113 |
Commutative algebra | 116 |
A return to finite type | 121 |
The set of finite type points is open | 126 |
Kohns algorithm for subelliptic multipliers | 129 |
Subelliptic estimates | 130 |
Kohns algorithm | 133 |
Kohns algorithm for holomorphic and formal germs | 134 |
Failure of effectiveness for Kohns algorithm | 139 |
Triangular systems | 140 |
Additional remarks | 144 |
Connections with partial differential equations | 147 |
Local regularity for | 149 |
Hypoellipticity global regularity and compactness | 150 |
An introduction to L2estimates | 152 |
Positivity conditions | 157 |
The classes Pk | 158 |
Intermediate conditions | 159 |
The global CauchySchwarz inequality | 161 |
A complicated example | 164 |
Stabilization in the bihomogeneous polynomial case | 166 |
Holomorphic Morse inequalities | 310 |
Subadditivity of multiplier ideals and Fujitas approximate Zariski | 329 |
Structure of the pseudoeffective cone and mobile intersection theory | 343 |
Supercanonical metrics and abundance | 357 |
Sius analytic approach and Păuns non vanishing theorem | 365 |
Introduction | 373 |
Proof of ii | 390 |
References | 402 |
Resolutions and principalizations | 409 |
Marked ideals | 415 |
Hypersurfaces of maximal contact and coefficient ideals | 423 |
Homogenized ideals | 431 |
Proof of principalization | 437 |
Bibliography | 449 |
Construction and examples of multiplier ideals | 455 |
Vanishing theorems for multiplier ideals | 463 |
Local properties of multiplier ideals | 471 |
Asymptotic constructions | 479 |
Extension theorems and deformation invariance of plurigenera | 487 |
Bibliography | 493 |
The cone of curves | 503 |
Flips | 513 |
Bibliography | 523 |
Plflips | 531 |
Finite generation of the restricted algebra | 545 |
Solutions to the exercises | 551 |
Alessio Corti Paul Hacking János Kollár Robert Lazarsfeld | 557 |
Existence of flips I | 565 |
Existence of flips II | 571 |
Bibliography | 583 |
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Analytic and Algebraic Geometry: Common Problems, Different Methods Jeffery D. McNeal,Mircea Mustata No preview available - 2010 |
Common terms and phrases
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