Vector bundle

About: Vector bundle is a research topic. Over the lifetime, 8517 publications have been published within this topic receiving 148965 citations.

Introduction to Smooth Manifolds

Abstract: Preface.- 1 Smooth Manifolds.- 2 Smooth Maps.- 3 Tangent Vectors.- 4 Submersions, Immersions, and Embeddings.- 5 Submanifolds.- 6 Sard's Theorem.- 7 Lie Groups.- 8 Vector Fields.- 9 Integral Curves and Flows.- 10 Vector Bundles.- 11 The Cotangent Bundle.- 12 Tensors.- 13 Riemannian Metrics.- 14 Differential Forms.- 15 Orientations.- 16 Integration on Manifolds.- 17 De Rham Cohomology.- 18 The de Rham Theorem.- 19 Distributions and Foliations.- 20 The Exponential Map.- 21 Quotient Manifolds.- 22 Symplectic Manifolds.- Appendix A: Review of Topology.- Appendix B: Review of Linear Algebra.- Appendix C: Review of Calculus.- Appendix D: Review of Differential Equations.- References.- Notation Index.- Subject Index

Compact Complex Surfaces

Abstract: Historical Note.- References.- The Content of the Book.- Standard Notations.- I. Preliminaries.- Topology and Algebra.- 1. Notations and Basic Facts.- 2. Some Properties of Bilinear forms.- 3. Vector Bundles, Characteristic Classes and the Index Theorem.- Complex Manifolds.- 4. Basic Concepts and Facts.- 5. Holomorphic Vector Bundles, Serre Duality and the Riemann-Roch Theorem.- 6. Line Bundles and Divisors.- 7. Algebraic Dimension and Kodaira Dimension.- General Analytic Geometry.- 8. Complex Spaces.- 9. The ?-Process.- 10. Deformations of Complex Manifolds.- Differential Geometry of Complex Manifolds.- 11. De Rham Cohomology.- 12. Dolbeault Cohomology.- 13. Kahler Manifolds.- 14. Weight-1 Hodge Structures.- 15. Yau's Results on Kahler-Einstein Metrics.- Coverings.- 16. Ramification.- 17. Cyclic Coverings.- 18. Covering Tricks.- Projective-Algebraic Varieties.- 19. GAGA Theorems and Projectivity Criteria.- 20. Theorems of Bertini and Lefschetz.- II. Curves on Surfaces.- Embedded Curves.- 1. Some Standard Exact Sequences.- 2. The Picard-Group of an Embedded Curve.- 3. Riemann-Roch for an Embedded Curve.- 4. The Residue Theorem.- 5. The Trace Map.- 6. Serre Duality on an Embedded Curve.- 7. The ?-Process.- 8. Simple Singularities of Curves.- Intersection Theory.- 9. Intersection Multiplicities.- 10. Intersection Numbers.- 11. The Arithmetical Genus of an Embedded Curve.- 12. 1-Connected Divisors.- III. Mappings of Surfaces.- Bimeromorphic Geometry.- 1. Bimeromorphic Maps.- 2. Exceptional Curves.- 3. Rational Singularities.- 4. Exceptional Curves of the First Kind.- 5. Hirzebruch-Jung Singularities.- 6. Resolution of Surface Singularities.- 7. Singularities of Double Coverings, Simple Singularities of Surfaces.- Fibrations of Surfaces.- 8. Generalities on Fibrations.- 9. The n-th Root Fibration.- 10. Stable Fibrations.- 11. Direct Image Sheaves.- 12. Relative Duality.- The Period Map of Stable Fibrations.- 13. Period Matrices of Stable Curves.- 14. Topological Monodromy of Stable Fibrations.- 15. Monodromy of the Period Matrix.- 16. Extending the Period Map.- 17. The Degree of f* ?X/S.- 18. Iitaka's Conjecture C2, 1.- IV. Some General Properties of Surfaces.- 1. Meromorphic Maps Associated to Line Bundles.- 2. Hodge Theory on Surfaces.- 3. Deformations of Surfaces.- 4. Some Inequahties for Hodge Numbers.- 5. Projectivity of Surfaces.- 6. Surfaces of Algebraic Dimension Zero.- 7. Almost-Complex Surfaces without any Complex Structure.- 8. The Vanishing Theorems of Ramanujam and Mumford.- V. Examples.- Some Classical Examples.- 1. The Projective Plane ?2.- 2. Complete Intersections.- 3. Tori of Dimension 2.- Fibre Bundles.- 4. Ruled Surfaces.- 5. Elliptic Fibre Bundles.- 6. Higher Genus Fibre Bundles.- Elliptic Fibrations.- 7. Kodaira's Table of Singular Fibres.- 8. Stable Fibrations.- 9. The Jacobian Fibration.- 10. Stable Reduction.- 11. Classification.- 12. Invariants.- 13. Logarithmic Transformations.- Kodaira Fibrations.- 14. Kodaira Fibrations.- Finite Quotients.- 15. The Godeaux Surface.- 16. Kummer Surfaces.- 17. Quotients of Products of Curves.- Infinite Quotients.- 18. Hopf Surfaces.- 19. Inoue Surfaces.- 20. Quotients of Bounded Domains in C2.- 21. Hilbert Modular Surfaces.- Double Coverings.- 22. Invariants.- 23. An Enriques Surface.- VI. The Enriques-Kodaira Classification.- 1. Statement of the Main Result.- 2. The Castelnuovo Criterion.- 3. The Case a(X) = 2.- 4. The Case a(X) = 1.- 5. The Case a (X) = 0.- 6. The Final Step.- 7. Deformations.- VII. Surfaces of General Type.- Preliminaries.- 1. Introduction.- 2. Some General Theorems.- Two Inequalities.- 3. Noether's Inequality.- 4. The Inequality c12 ? 3c2.- Pluricanonical Maps.- 5. The Main Results.- 6. Connectedness Properties of Pluricanonical Divisors.- 7. Proof of the Main Results.- 8. The Exceptional Cases and the 1-canonical Map.- Surfaces with Given Chern Numbers.- 9. The Geography of Chern Numbers.- 10. Surfaces on the Noether Lines.- 11. Surfaces with q = pg = 0.- VIII. K3-Surfaces and Enriques Surfaces.- 1. Notations.- 2. The Results.- K3-Surfaces.- 3. Topological and Analytical Invariants.- 4. Digression on Affine Geometry over ?2.- 5. The Picard Lattice of Kummer Surfaces.- 6. The Torelli Theorem for Kummer Surfaces.- 7. The Local Torelli Theorem for K3-Surfaces.- 8. A Density Theorem.- 9. Behaviour of the Kahler Cone Under Deformations.- 10. Degenerations of Isomorphisms Between Kahler K3-Surfaces.- 11. The Torelli Theorems for Kahler K3-Surfaces.- 12. Construction of Moduli Spaces.- 13. Digression on Quaternionic Structures.- 14. Surjectivity of the Period Map Every K3-Surface is Kahlerian.- Enriques Surfaces.- 15. Topological and Analytic Invariants.- 16. Divisors on an Enriques Surface Y.- 17. Elliptic Pencils.- 18. Double Coverings of Quadrics.- 19. The Period Map.- 20. The Period Domain for Enriques Surfaces.- 21. Global Properties of the Period Map.- Notations.

Heat Kernels and Dirac Operators

Abstract: 1 Background on Differential Geometry.- 1.1 Fibre Bundles and Connections.- 1.2 Riemannian Manifolds.- 1.3 Superspaces.- 1.4 Superconnections.- 1.5 Characteristic Classes.- 1.6 The Euler and Thorn Classes.- 2 Asymptotic Expansion of the Heat Kernel.- 2.1 Differential Operators.- 2.2 The Heat Kernel on Euclidean Space.- 2.3 Heat Kernels.- 2.4 Construction of the Heat Kernel.- 2.5 The Formal Solution.- 2.6 The Trace of the Heat Kernel.- 2.7 Heat Kernels Depending on a Parameter.- 3 Clifford Modules and Dirac Operators.- 3.1 The Clifford Algebra.- 3.2 Spinors.- 3.3 Dirac Operators.- 3.4 Index of Dirac Operators.- 3.5 The Lichnerowicz Formula.- 3.6 Some Examples of Clifford Modules.- 4 Index Density of Dirac Operators.- 4.1 The Local Index Theorem.- 4.2 Mehler's Formula.- 4.3 Calculation of the Index Density.- 5 The Exponential Map and the Index Density.- 5.1 Jacobian of the Exponential Map on Principal Bundles.- 5.2 The Heat Kernel of a Principal Bundle.- 5.3 Calculus with Grassmann and Clifford Variables.- 5.4 The Index of Dirac Operators.- 6 The Equivariant Index Theorem.- 6.1 The Equivariant Index of Dirac Operators.- 6.2 The Atiyah-Bott Fixed Point Formula.- 6.3 Asymptotic Expansion of the Equivariant Heat Kernel.- 6.4 The Local Equivariant Index Theorem.- 6.5 Geodesic Distance on a Principal Bundle.- 6.6 The heat kernel of an equivariant vector bundle.- 6.7 Proof of Proposition 6.13.- 7 Equivariant Differential Forms.- 7.1 Equivariant Characteristic Classes.- 7.2 The Localization Formula.- 7.3 Bott's Formulas for Characteristic Numbers.- 7.4 Exact Stationary Phase Approximation.- 7.5 The Fourier Transform of Coadjoint Orbits.- 7.6 Equivariant Cohomology and Families.- 7.7 The Bott Class.- 8 The Kirillov Formula for the Equivariant Index.- 8.1 The Kirillov Formula.- 8.2 The Weyl and Kirillov Character Formulas.- 8.3 The Heat Kernel Proof of the Kirillov Formula.- 9 The Index Bundle.- 9.1 The Index Bundle in Finite Dimensions.- 9.2 The Index Bundle of a Family of Dirac Operators.- 9.3 The Chern Character of the Index Bundle.- 9.4 The Equivariant Index and the Index Bundle.- 9.5 The Case of Varying Dimension.- 9.6 The Zeta-Function of a Laplacian.- 9.7 The Determinant Line Bundle.- 10 The Family Index Theorem.- 10.1 Riemannian Fibre Bundles.- 10.2 Clifford Modules on Fibre Bundles.- 10.3 The Bismut Superconnection.- 10.4 The Family Index Density.- 10.5 The Transgression Formula.- 10.6 The Curvature of the Determinant Line Bundle.- 10.7 The Kirillov Formula and Bismut's Index Theorem.- References.- List of Notation.

Stable mappings and their singularities

Abstract: I: Preliminaries on Manifolds.- 1. Manifolds.- 2. Differentiable Mappings and Submanifolds.- 3. Tangent Spaces.- 4. Partitions of Unity.- 5. Vector Bundles.- 6. Integration of Vector Fields.- II: Transversality.- 1. Sard's Theorem.- 2. Jet Bundles.- 3. The Whitney C? Topology.- 4. Transversality.- 5. The Whitney Embedding Theorem.- 6. Morse Theory.- 7. The Tubular Neighborhood Theorem.- III: Stable Mappings.- 1. Stable and Infinitesimally Stable Mappings.- 2. Examples.- 3. Immersions with Normal Crossings.- 4. Submersions with Folds.- IV: The Malgrange Preparation Theorem.- 1. The Weierstrass Preparation Theorem.- 2. The Malgrange Preparation Theorem.- 3. The Generalized Malgrange Preparation Theorem.- V: Various Equivalent Notions of Stability.- 1. Another Formulation of Infinitesimal Stability.- 2. Stability Under Deformations.- 3. A Characterization of Trivial Deformations.- 4. Infinitesimal Stability => Stability.- 5. Local Transverse Stability.- 6. Transverse Stability.- 7. Summary.- VI: Classification of Singularities, Part I: The Thom-Boardman Invariants.- 1. The Sr Classification.- 2. The Whitney Theory for Generic Mappings between 2-Manifolds.- 3. The Intrinsic Derivative.- 4. The Sr,s Singularities.- 5. The Thom-Boardman Stratification.- 6. Stable Maps Are Not Dense.- VII: Classification of Singularities, Part II: The Local Ring of a Singularity.- 1. Introduction.- 2. Finite Mappings.- 3. Contact Classes and Morin Singularities.- 4. Canonical Forms for Morin Singularities.- 5. Umbilics.- 6. Stable Mappings in Low Dimensions.- A. Lie Groups.- Symbol Index.

Topological methods in algebraic geometry

Abstract: Introduction Chapter 1: Preparatory material 1. Multiplicative sequences 2. Sheaves 3. Fibre bundles 4. Characteristic classes Chapter 2: The cobordism ring 5. Pontrjagin numbers 6. The ring /ss(/Omega) /oplus //Varrho 7. The cobordism ring /omega 8. The index of a 4k-dimensional manifold 9. The virtual index Chapter 3: The Todd genus 10. Definiton of the Todd genus 11. The virutal generalised Todd genus 12. The t-characteristic of a GL(q, C)-bundle 13. Split manifolds and splitting methods 14. Multiplicative properties of the Todd genus Chapter 4: The Riemann-Roch theorem for algebraic manifolds 15. Cohomology of Compact complex manifolds 16. Further properties of the (/chi)x characteristics 17. The virtual (/chi)x characteristics 18. Some fundamental theorems of Kodaira 19. The virtual (/chi)x characteristics for algebraic manifolds 20. The Riemann-Roch theorem for algebraic manifolds and complex analytic line bundles 21. The Riemann-Roch theorem for algebraic manifolds and complex analytic vector bundles Appendix 1 by R.L.E. Schwarzenberger 22. Applications of the Riemann-Roch theorem 23. The Riemann-Roch theorem of Grothendieck 24. The Grothendieck ring of continuous vector bundles 25. The Atijah-Singer index theorem 26. Integrality theorems for differentiable manifolds Appendix 2 by A. Borel A spectral sequence for complex analytic bundles Bibliography Index