Holdings: Principles of algebraic geometry

Principles of algebraic geometry:

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Bibliographic Details
Main Authors:	Griffiths, Phillip 1938- (Author), Harris, Joseph (Author)
Format:	Book
Language:	English
Published:	New York [u.a.] Wiley 1994
Series:	Wiley classics library A Wiley interscience publication
Subjects:	Algebraische Geometrie
Online Access:	Inhaltsverzeichnis
Physical Description:	XII, 813 S. graph. Darst.
ISBN:	0471050598 9780471050599

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MARC


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adam_text	CONTENTS CHAPTER 0 FOUNDATIONAL MATERIAL 1 1. Rudiments of Several Complex Variables Cauchy s Formula and Applications 2 Several Variables 6 Weierstrass Theorems and Corollaries 7 Analytic Varieties 12 2. Complex Manifolds Complex Manifolds 14 Submanifolds and Subvarieties 18 De Rham and Dolbeault Cohomology 23 Calculus on Complex Manifolds 27 3. Sheaves and Cohomology Origins: The Mittag-Leffler Problem 34 Sheaves 35 Cohomology of Sheaves 38 The de Rham Theorem 43 The Dolbeault Theorem 45 4. Topology of Manifolds Intersection of Cycles 49 Poincaré Duality 53 Intersection of Analytic Cycles 60 5. Vector Bundles, Connections, and Curvature Complex and Holomorphic Vector Bundles 66 Metrics, Connections, and Curvature 71 6. Harmonic Theory on Compact Complex Manifolds The Hodge Theorem 80 Proof of the Hodge Theorem I: Local Theory 84 vii ущ CONTENTS Proof of the Hodge Theorem II: Global Theory 92 Applications of the Hodge Theorem 100 7. Kahler Manifolds The Kahler Condition 106 The Hodge Identities and the Hodge Decomposition 111 The Lef schetz Decomposition 118 CHAPTER 1 COMPLEX ALGEBRAIC VARIETIES 128 1. Divisors and Line Bundles Divisors 129 Line Bundles I32 Chera Classes of Line Bundles 139 2. Some Vanishing Theorems and Corollaries The Kodaira Vanishing Theorem 148 The Lef schetz Theorem on Hyperplane Sections 156 Theorem В І59 The Lefschetz Theorem on (1, l)-classes 161 3. Algebraic Varieties Analytic and Algebraic Varieties 164 Degree of a Variety »71 Tangent Spaces to Algebraic Varieties 175 4. The Kodaira Embedding Theorem Line Bundles and Maps to Projective Space 176 Blowing Up 182 Proof of the Kodaira Theorem 189 5. Grassmannians Definitions 193 The Cell Decomposition 194 The Schubert Calculus 197 Universal Bundles 207 The Pliicker Embedding 209 CHAPTER 2 RIEMANN SURFACES AND ALGEBRAIC CURVES 212 1. Preliminaries Embedding Riemann Surfaces 213 The Riemann-Hurwitz Formula 216 CONTENTS ix The Genus Formula 219 Cases g =0,1 222 2. Abel s Theorem Abel s Theorem — First Version 224 The First Reciprocity Law and Corollaries 229 Abel s Theorem — Second Version 232 Jacobi Inversion 235 3. Linear Systems on Curves Reciprocity Law II 240 The Riemann-Roch Formula 243 Canonical Curves 246 Special Linear Systems I 249 Hyperelliptic Curves and Riemann s Count 253 Special Linear Systems II 259 4. Pliicker Formulas Associated Curves 263 Ramification 264 The General Plücker Formulas I 268 The General Plücker Formulas II 271 Weierstrass Points 273 Plücker Formulas for Plane Curves 277 5. Correspondences Definitions and Formulas 282 Geometry of Space Curves 290 Special Linear Systems III 298 6. Complex Tori and Abelian Varieties The Riemann Conditions 300 Line Bundles on Complex Tori 307 Theta-Functions 317 The Group Structure on an Abelian Variety 324 Intrinsic Formulations 326 7. Curves and Their Jacobians Preliminaries 333 Riemann s Theorem 338 Riemann s Singularity Theorem 341 Special Linear Systems IV 349 Torelli s Theorem 359 x CONTENTS CHAPTER 3 FURTHER TECHNIQUES 364 1. Distributions and Currents Definitions; Residue Formulas 366 Smoothing and Regularity 373 Cohomology of Currents 382 2. Applications of Currents to Complex Analysis Currents Associated to Analytic Varieties 385 Intersection Numbers of Analytic Varieties 392 The Levi Extension and Proper Mapping Theorems 395 3. Chern Classes Definitions 400 The Gauss Bonnet Formulas 409 Some Remarks — Not Indispensable — Concerning Chern Classes of Holomorphic Vector Bundles 416 4. Fixed-Point and Residue Formulas The Lefschetz Fixed-Point Formula 419 The Holomorphic Lefschetz Fixed-Point Formula 422 The Bott Residue Formula 426 The General Hirzebruch-Riemann-Roch Formula 435 5. Spectral Sequences and Applications Spectral Sequences of Filtered and Bigraded Complexes 438 Hypercohomology 445 Differentials of the Second Kind 454 The Leray Spectral Sequence 462 CHAPTER 4 SURFACES 469 1. Preliminaries Intersection Numbers, the Adjunction Formula, and Riemann-Roch 470 Blowing Up and Down 473 The Quadric Surface 478 The Cubic Surface 480 2. Rational Maps Rational and Birational Maps 489 Curves on an Algebraic Surface 498 The Structure of Birational Maps Between Surfaces 510 3. Rational Surfaces I Noether s Lemma 513 Rational Ruled Surfaces 514 CONTENTS xi The General Rational Surface 520 Surfaces of Minimal Degree 522 Curves of Maximal Genus 527 Steiner Constructions 528 The Enriques-Petri Theorem 533 4. Rational Surfaces II The Castelnuovo-Enriques Theorem 536 The Enriques Surface 541 Cubic Surfaces Revisited 545 The Intersection of Two Quadrics in P4 550 5. Some Irrational Surfaces The Albanese Map 552 Irrational Ruled Surfaces 553 A Brief Introduction to Elliptic Surfaces 564 Kodaira Number and the Classification Theorem I 572 The Classification Theorem II 582 K-3 Surfaces 590 Enriques Surfaces 594 6. Noether s Formula Noether s Formula for Smooth Hypersurfaces 600 Blowing Up Submanifolds 602 Ordinary Singularities of Surfaces 611 Noether s Formula for General Surfaces 618 Some Examples 628 Isolated Singularities of Surfaces 636 CHAPTER 5 RESIDUES 647 1. Elementary Properties of Residues Definition and Cohomological Interpretation 649 The Global Residue Theorem 655 The Transformation Law and Local Duality 656 2. Applications of Residues Intersection Numbers 662 Finite Holomorphic Mappings 667 Applications to Plane Projective Geometry 670 3. Rudiments of Commutative and Homological Algebra with Applications Commutative Algebra 678 Homological Algebra 682 The Koszul Complex and Applications 687 A Brief Tour Through Coherent Sheaves 695 xli CONTENTS 4. Global Duality Global Ext 705 Explanation of the General Global Duality Theorem 707 Global Ext and Vector Fields with Isolated Zeros 708 Global Duality and Superabundance of Points on a Surface 712 Extensions of Modules 722 Points on a Surface and Rank-Two Vector Bundles 726 Residues and Vector Bundles 729 CHAPTER б THE QUADRIC LINE COMPLEX 733 1. Preliminaries: Quadrics Rank of a Quadric 734 Linear Spaces on Quadrics 735 Linear Systems of Quadrics 741 Lines on Linear Systems of Quadrics 746 The Problem of Five Conies 749 2. The Quadric Line Complex: Introduction Geometry of the Grassmannian G(2,4) 756 Line Complexes 759 The Quadric Line Complex and Associated Kummer Surface I 762 Singular Lines of the Quadric Line Complex 767 Two Configurations 773 3. Lines on the Quadric Line Complex The Variety of Lines on the Quadric Line Complex 778 Curves on the Variety of Lines 780 Two Configurations Revisited 784 The Group Law 787 4. The Quadric Line Complex: Reprise The Quadric Line Complex and Associated Kummer Surface II 791 Rationality of the Quadric Line Complex 796 INDEX 805
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indexdate	2024-07-09T17:48:37Z
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isbn	0471050598 9780471050599
language	English
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series2	Wiley classics library A Wiley interscience publication
spelling	Griffiths, Phillip 1938- Verfasser (DE-588)131881434 aut Principles of algebraic geometry Phillip Griffiths and Joseph Harris New York [u.a.] Wiley 1994 XII, 813 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Wiley classics library A Wiley interscience publication Algebraische Geometrie (DE-588)4001161-6 gnd rswk-swf Algebraische Geometrie (DE-588)4001161-6 s DE-604 Harris, Joseph Verfasser aut Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006788602&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Griffiths, Phillip 1938- Harris, Joseph Principles of algebraic geometry Algebraische Geometrie (DE-588)4001161-6 gnd
subject_GND	(DE-588)4001161-6
title	Principles of algebraic geometry
title_auth	Principles of algebraic geometry
title_exact_search	Principles of algebraic geometry
title_full	Principles of algebraic geometry Phillip Griffiths and Joseph Harris
title_fullStr	Principles of algebraic geometry Phillip Griffiths and Joseph Harris
title_full_unstemmed	Principles of algebraic geometry Phillip Griffiths and Joseph Harris
title_short	Principles of algebraic geometry
title_sort	principles of algebraic geometry
topic	Algebraische Geometrie (DE-588)4001161-6 gnd
topic_facet	Algebraische Geometrie
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006788602&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
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