Topic
Fibration
About: Fibration is a research topic. Over the lifetime, 3529 publications have been published within this topic receiving 52168 citations.
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Book•
1 Jan 1984
TL;DR: In this article, the authors describe the topology and algebraic properties of complex surfaces, including the following properties: 1. The Projective Plane, 2. The Jacobian Fibration, 3. Hodge Theory on Surfaces, 4. Inequahties for Hodge Numbers, 5. Holomorphic Vector Bundles, Serre Duality and Riemann-Roch Theorem.
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.
3,036 citations
TL;DR: In this paper, the authors constructed compact examples of D-manifolds for type IIB strings and showed that the construction has a natural interpretation in terms of compactification of a 12-dimensional ''F-theory''.
2,035 citations
TL;DR: In this paper, the De Rham Complex of a Compact K~ihler Manifold has been shown to be a homotopy theory of differential algebras.
Abstract: 1. Homotopy Theory of Differential Algebras . . . . . . . . . . 248 2. De Rham Homotopy Theory . . . . . . . . . . . . . . . 254 3. Relation between De Rham Homotopy Theory and Classical Homotopy Theory . . . . . . . . . . . . . . . . . . . . . . . . 256 4. Formality of Differential Algebras . . . . . . . . . . . . . . 260 5. The De Rham Complex of a Compact K~ihler Manifold . . . . . 262 6. The Main Theorem and Two Proofs . . . . . . . . . . . . . 270 7. An Application . . . . . . . . . . . . . . . . . . . . . . 272
1,083 citations
TL;DR: In this paper, the deformation theory necessary to obtain virtual moduli cycles of stable sheaves whose higher obstruction groups vanish has been developed, and the moduli spaces of sheaves on a general $K3$ fibration have been computed.
Abstract: We briefly review the formal picture in which a Calabi-Yau $n$-fold is the complex analogue of an oriented real $n$-manifold, and a Fano with a fixed smooth anticanonical divisor is the analogue of a manifold with boundary, motivating a holomorphic Casson invariant counting bundles on a Calabi-Yau 3-fold. We develop the deformation theory necessary to obtain the virtual moduli cycles of \cite{LT}, \cite{BF} in moduli spaces of stable sheaves whose higher obstruction groups vanish. This gives, for instance, virtual moduli cycles in Hilbert schemes of curves in $\Pee^3$, and Donaldson-- and Gromov-Witten-- like invariants of Fano 3-folds. It also allows us to define the holomorphic Casson invariant of a Calabi-Yau 3-fold $X$, prove it is deformation invariant, and compute it explicitly in some examples. Then we calculate moduli spaces of sheaves on a general $K3$ fibration $X$, enabling us to compute the invariant for some ranks and Chern classes, and equate it to Gromov-Witten invariants of the ``Mukai-dual'' 3-fold for others. As an example the invariant is shown to distinguish Gross' diffeomorphic 3-folds. Finally the Mukai-dual 3-fold is shown to be Calabi-Yau and its cohomology is related to that of $X$.
464 citations
Book•
1 Jan 1981
TL;DR: In this paper, Quillen's work on Rational Homotopy Theory of DGAs and rational homotopy theory for simplicial complexes has been studied in the context of structural analysis.
Abstract: 1 Introduction- 2 Basic Concepts- 3 CW Homology Theorem- 4 The Whitehead Theorem and the Hurewicz Theorem- 5 Spectral Sequence of a Fibration- 6 Obstruction Theory- 7 Eilenberg-MacLane Spaces, Cohomology, and Principal Fibrations- 8 Postnikov Towers and Rational Homotopy Theory- 9 deRham's theorem for simplicial complexes- 10 Differential Graded Algebras- 11 Homotopy Theory of DGAs- 12 DGAs and Rational Homotopy Theory- 13 The Fundamental Group- 14 Examples and Computations- 15 Functorality- 16 The Hirsch Lemma- 17 Quillen's work on Rational Homotopy Theory- 18 A1-structures and C1-structures- 19 Exercises
344 citations
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| Year | Papers |
|---|---|
| 2026 | 3 |
| 2025 | 47 |
| 2024 | 43 |
| 2023 | 170 |
| 2022 | 426 |
| 2021 | 190 |