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Intro; Preface; References; Contents; Symbols for "Abelian and Triangulated Categories"; Part I Derived Categories and Related Topics in Algebraic Geometry; Abelian and Triangulated Categories; 1 Preface; 2 Introduction; 3 Additive Categories; 3.1 Definition of Additive Categories; 3.2 k-Linear Categories; 3.3 Additive Functors; 4 Abelian Categories; 4.1 Some Category Theory; 4.2 Definition of Abelian Categories; 4.3 Examples; 5 Triangulated Categories; 5.1 Some More Category Theory; 5.2 Definition of Triangulated Categories; 5.3 Axioms; 5.4 Results about Triangulated Categories; References
Derived Categories and Derived Functors1 Category of Complexes and Homotopy Category; 2 Defining the Derived Category; 3 Derived Category as a Triangulated Category; 4 Equivalence Between Homotopy Category of Injectives in mathcalA and the Derived Category D(mathcalA); 5 Derived Functors; References; Introduction to Quivers; 1 Quivers and Path Algebras; 2 The Category of Quiver Representations; 3 Bound Quivers; References; Semi-orthogonal Decompositions of Derived Categories; 1 Preliminaries; 2 Two Classes of Examples; 2.1 Derived Categories of Bound Quivers Db(Q,I)
2.2 Derived Categories of Coherent Sheaves Db(X)2.3 Actions on SODs; 2.4 More Examples; 3 Connections; References; Introduction to Stability Conditions; 1 Motivation; 2 Definition of Stability; 2.1 Example: mathbbP1; 2.2 Definition; 3 Examples; 3.1 Curves; 3.2 Quivers; 3.3 Surfaces, Threefolds, and Higher Dimensional Varieties; References; A Brief Introduction to Geometric Invariant Theory; 1 Introduction; 2 Reductive Groups; 3 Group Quotients; 4 Linearization of an Invertible Sheaf; 5 Semi-Stability and the First Main Theorem of GIT; 6 The Numerical Criterion
7 Stratifying the Unstable LocusReferences; Birational Geometry and Derived Categories; References; Part II Approaches to Mirror Symmetry; Introduction to Mirror Symmetry; 1 A Guide to the References; 2 Review of Differential Forms and Complex Geometry; 2.1 Differential Topology; 2.2 Complex Geometry; 3 Kähler Geometry; 3.1 Definition and Examples; 3.2 Hodge Theory; 4 Mirror Symmetry and Calabi-Yau Manifolds; 4.1 Statement of Mirror Symmetry; 4.2 Calabi-Yau Manifolds; 4.3 Mirror Symmetry for Simply Connected Calabi-Yau 3-Folds; 5 Canonical Example: The Quintic Threefold; References
Batyrev Mirror Symmetry1 Introduction; 2 Polar Duality of Lattice Polytopes; 3 Varieties from Lattice Polytopes; 4 Batyrev's Construction; 5 The Quintic Threefold; 6 Further Developments; References; Introduction to Differential Graded Categories; 1 Motivation; 2 Definitions; 2.1 dg Categories; 2.2 The dg Category of R-Modules; 3 Triangulated Structures and dg Categories; 4 Functor Categories and Modules over dg Categories; 4.1 The Category of dg Categories; 4.2 Modules over dg Categories; 4.3 The Yoneda Embedding; 4.4 The dg Derived Category; 5 Additive Invariants
Derived Categories and Derived Functors1 Category of Complexes and Homotopy Category; 2 Defining the Derived Category; 3 Derived Category as a Triangulated Category; 4 Equivalence Between Homotopy Category of Injectives in mathcalA and the Derived Category D(mathcalA); 5 Derived Functors; References; Introduction to Quivers; 1 Quivers and Path Algebras; 2 The Category of Quiver Representations; 3 Bound Quivers; References; Semi-orthogonal Decompositions of Derived Categories; 1 Preliminaries; 2 Two Classes of Examples; 2.1 Derived Categories of Bound Quivers Db(Q,I)
2.2 Derived Categories of Coherent Sheaves Db(X)2.3 Actions on SODs; 2.4 More Examples; 3 Connections; References; Introduction to Stability Conditions; 1 Motivation; 2 Definition of Stability; 2.1 Example: mathbbP1; 2.2 Definition; 3 Examples; 3.1 Curves; 3.2 Quivers; 3.3 Surfaces, Threefolds, and Higher Dimensional Varieties; References; A Brief Introduction to Geometric Invariant Theory; 1 Introduction; 2 Reductive Groups; 3 Group Quotients; 4 Linearization of an Invertible Sheaf; 5 Semi-Stability and the First Main Theorem of GIT; 6 The Numerical Criterion
7 Stratifying the Unstable LocusReferences; Birational Geometry and Derived Categories; References; Part II Approaches to Mirror Symmetry; Introduction to Mirror Symmetry; 1 A Guide to the References; 2 Review of Differential Forms and Complex Geometry; 2.1 Differential Topology; 2.2 Complex Geometry; 3 Kähler Geometry; 3.1 Definition and Examples; 3.2 Hodge Theory; 4 Mirror Symmetry and Calabi-Yau Manifolds; 4.1 Statement of Mirror Symmetry; 4.2 Calabi-Yau Manifolds; 4.3 Mirror Symmetry for Simply Connected Calabi-Yau 3-Folds; 5 Canonical Example: The Quintic Threefold; References
Batyrev Mirror Symmetry1 Introduction; 2 Polar Duality of Lattice Polytopes; 3 Varieties from Lattice Polytopes; 4 Batyrev's Construction; 5 The Quintic Threefold; 6 Further Developments; References; Introduction to Differential Graded Categories; 1 Motivation; 2 Definitions; 2.1 dg Categories; 2.2 The dg Category of R-Modules; 3 Triangulated Structures and dg Categories; 4 Functor Categories and Modules over dg Categories; 4.1 The Category of dg Categories; 4.2 Modules over dg Categories; 4.3 The Yoneda Embedding; 4.4 The dg Derived Category; 5 Additive Invariants