Abelian category

About: Abelian category is a research topic. Over the lifetime, 1772 publications have been published within this topic receiving 36347 citations.

Complex Abelian varieties

Abstract: Notation- 1 Complex Tori- 2 Line Bundles on Complex Tori- 3 Cohomology of Line Bundles- 4 Abelian Varieties- 5 Endomorphisms of Abelian Varieties- 6 Theta and Heisenberg Groups- 7 Equations for Abelian Varieties- 8 Moduli- 9 Moduli Spaces of Abelian Varieties with Endomorphism Structure- 10 Abelian Surfaces- 11 Jacobian Varieties- 12 Prym Varieties- 13 Automorphisms- 14 Vector bundles on Abelian Varieties- 15 Further Results on Line Bundles an the Theta Divisor- 16 Cycles on Abelian varieties- 17 The Hodge Conjecture for General Abelian and Jacobian Varieties- A Algebraic Varieties and Complex Analytic Spaces- B Line Bundles and Factors of Automorphy- C Some Algebraic Geometric Results- C1 Some Properties of ?-Divisors- C2 The Kodaira Dimension- C3 Vanishing Theorems- C6 Some Results from Intersection Theory- C7 Adjoint Ideals- D Derived Categories- D1 Definition and First Properties- D2 Derived Functors- D3 The Grothendieck-Riemann-Roch Theorem- E Moduli Spaces of Sheaves- F Abelian Schemes- F1 Abelian Schemes and the Poincar'e Bundle- F2 Relative Fourier Functor- F3 The Relative Jacobian- Glossary of Notation

Fourier-Mukai transforms in algebraic geometry

Abstract: Preface 1. Triangulated categories 2. Derived categories: a quick tour 3. Derived categories of coherent sheaves 4. Derived category and canonical bundle I 5. Fourier-Mukai transforms 6. Derived category and canonical bundle II 7. Equivalence criteria for Fourier-Mukai transforms 8. Spherical and exceptional objects 9. Abelian varieties 10. K3 surfaces 11. Flips and flops 12. Derived categories of surfaces 13. Where to go from here References Index

On fusion categories

Abstract: Using a variety of methods developed in the literature (in particular, the theory of weak Hopf algebras), we prove a number of general results about fusion categories in characteristic zero. We show that the global dimension of a fusion category is always positive, and that the S-matrix of any (not necessarily hermitian) modular category is unitary. We also show that the category of module functors between two module categories over a fusion category is semisimple, and that fusion categories and tensor functors between them are undeformable (generalized Ocneanu rigidity). In particular the number of such categories (functors) realizing a given fusion datum is finite. Finally, we develop the theory of Frobenius-Perron dimensions in an arbitrary fusion category. At the end of the paper we generalize some of these results to positive characteristic.

Categories and Sheaves

Abstract: The Language of Categories.- Limits.- Filtrant Limits.- Tensor Categories.- Generators and Representability.- Indization of Categories.- Localization.- Additive and Abelian Categories.- ?-accessible Objects and F-injective Objects.- Triangulated Categories.- Complexes in Additive Categories.- Complexes in Abelian Categories.- Derived Categories.- Unbounded Derived Categories.- Indization and Derivation of Abelian Categories.- Grothendieck Topologies.- Sheaves on Grothendieck Topologies.- Abelian Sheaves.- Stacks and Twisted Sheaves.