Kooperativer Bibliotheksverbund

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Format: 1 online resource (318 pages)

Edition: 1st ed.

ISBN: 9783031617058

Series Statement: Lecture Notes in Mathematics Series ; v.2350

Note: Intro -- Preface -- Acknowledgements -- Contents -- 1 Introduction -- 1.1 Motivation -- 1.1.1 Background -- 1.1.2 Categorical DT Theory -- 1.1.3 The Main Player in This Book -- 1.1.4 Why Do We Want to Categorify? -- 1.1.5 Enumerative Geometry vs Category Theory -- 1.1.6 Motivation from d-Critical Birational Geometry -- 1.1.7 Further Motivations -- 1.2 Categorical DT Theory for Local Surfaces -- 1.2.1 (-1)-Shifted Cotangent -- 1.2.2 Idea from Koszul Duality -- 1.2.3 DT Category for Stable Sheaves -- 1.2.4 De-categorification of DT Categories -- 1.3 D/K Conjectures in Categorical DT Theory -- 1.3.1 DT Categories of One-Dimensional Sheaves -- 1.3.2 Categorical MNOP/PT Theories -- 1.3.3 DT Category for Stable D0-D2-D6 Bound States -- 1.4 Three Approaches -- 1.4.1 Wall-Crossing via Linear Koszul Duality -- 1.4.2 Window Theorem for DT Categories -- 1.4.3 Categorified Hall Products -- 1.5 Updates of the Developments -- 1.5.1 The Z/2-Periodic Version -- 1.5.2 Window Subcategories via -Stacks -- 1.5.3 Categorical Wall-Crossing Formula -- 1.5.4 Applications to Derived Categories of Classical Moduli Spaces -- 1.5.5 Quasi-BPS Categories -- 1.6 Organization of This Book -- 1.6.1 Plan of the Book -- 1.6.2 Notation and Convention -- 2 Koszul Duality Equivalence -- 2.1 Singular Supports of Coherent Sheaves -- 2.1.1 Local Model -- 2.1.2 Definition of Singular Supports -- 2.1.3 Singular Supports via Relative Hochschild Cohomology -- 2.2 The Derived Categories of Factorizations -- 2.2.1 Definition of Factorizations -- 2.2.2 Functoriality of Derived Categories of Factorizations -- 2.3 Koszul Duality Equivalence -- 2.3.1 G-equivariant Tuple -- 2.3.2 Koszul Duality Equivalence -- 2.3.3 Singular Supports Under Koszul Duality -- 2.4 Some Functorial Properties of Koszul Duality Equivalence -- 2.4.1 Functoriality Under Push-Forward -- 2.4.2 Functoriality Under Pull-Back. , 3 Categorical DT Theory for Local Surfaces -- 3.1 Some Background on Derived Stacks -- 3.1.1 Quasi-Smooth Derived Stack -- 3.1.2 Ind-Coherent Sheaves on QCA Stacks -- 3.1.3 (-1)-Shifted Cotangent Derived Stack -- 3.1.4 Good Moduli Spaces of Artin Stacks -- 3.2 Categorical DT Theory via Verdier Quotients -- 3.2.1 Definition of DT Category -- 3.2.2 DT"0362DT-Version -- 3.2.3 Replacement of the Quotient Category -- 3.2.4 C-Rigidification -- 3.2.5 Functoriality of DT Categories -- 3.3 Comparison with Cohomological/Numerical DT Invariants -- 3.3.1 Periodic Cyclic Homologies (Review) -- 3.3.2 Periodic Cyclic Homologies for Derived Categories of Factorizations -- 3.3.3 Conjectural Relation with Cohomological DT Theory -- 3.3.4 Relation with Numerical DT Invariants -- 3.4 DT Categories for Local Surfaces -- 3.4.1 Derived Moduli Stacks of Coherent Sheaves on Surfaces -- 3.4.2 Moduli Stacks of Compactly Supported Sheaves on Local Surfaces -- 3.4.3 Definition of DT Category for Local Surfaces -- 4 D-Critical D/K Equivalence Conjectures -- 4.1 Categorical DT Theory for One Dimensional Stable Sheaves -- 4.1.1 Moduli Stacks of One Dimensional Stable Sheaves -- 4.1.2 Categorical Wall-Crossing for One Dimensional Sheaves -- 4.1.3 The Case of Irreducible Curve Class -- 4.2 Moduli Stacks of D0-D2-D6 Bound States -- 4.2.1 Moduli Stacks of Pairs -- 4.2.2 Moduli Stacks of D0-D2-D6 Bound States -- 4.3 DT Category for D0-D2-D6 Bound States -- 4.3.1 Categorical MNOP/PT Theories -- 4.3.2 Moduli Stacks of Stable D0-D2-D6 Bound States -- 4.3.3 DT Categories for Semistable D0-D2-D6 Bound States -- 4.3.4 Moduli Stacks of Semistable Pairs -- 4.3.5 Conjectural Wall-Crossing Phenomena of DT Categories -- 4.4 Example: Local (-1, -1)-Curve -- 4.4.1 Local (-1, -1)-Curve -- 4.4.2 Moduli Stacks of Quiver Representations -- 4.4.3 Moduli Stacks of Semistable Representations. , 4.4.4 Window Subcategories -- 4.4.5 Proof of Conjectures 4.3.3, 4.3.14 Local (-1, -1)-Curve -- 5 Categorical Wall-Crossing via Koszul Duality -- 5.1 Dualities of DT Categories for D0-D2-D6 Bound States -- 5.1.1 Moduli Stacks of Dual Pairs -- 5.1.2 Wall-Crossing at t< -- 0 -- 5.1.3 Wall-Crossing Formula of Categorical PT Theories for Irreducible Curve Classes -- 5.1.4 Application to the Rationality -- 5.2 The Category of D0-D2-D6 Bound States -- 5.2.1 Notation of Local Surfaces -- 5.2.2 The Category BS -- 5.2.3 The Functor from AX to BS -- 5.2.4 The Functor from BS to Dbcoh(X) -- 5.2.5 Moduli Stacks of Rank One Objects in AX -- 5.2.6 Comparison of Dualities -- 5.3 Semiorthogonal Decomposition via Koszul Duality -- 5.3.1 Linear Koszul Duality: Local Case -- 5.3.2 Linear Koszul Duality: Global Case -- 5.3.3 Semiorthogonal Decomposition -- 5.3.4 Singular Supports Under Linear Koszul Duality -- 6 Window Theorem for DT Categories -- 6.1 Window Theorem for GIT Quotient -- 6.1.1 Kempf-Ness Stratification -- 6.1.2 Semiorthogonal Decomposition via KN Stratification -- 6.1.3 Magic Window Subcategories -- 6.1.4 Window Subcategories for Formal Completions -- 6.1.5 KN Stratifications for Some Representations of Quivers -- 6.2 Intrinsic Window Subcategories -- 6.2.1 Symmetric Structures of Derived Stacks -- 6.2.2 Intrinsic Window Subcategories -- 6.3 Window Theorem for DT Categories -- 6.3.1 Window Subcategories Under Koszul Duality (Linear Case) -- 6.3.2 Window Subcategories Under Koszul Duality (Formal Fiber Case) -- 6.3.3 Window Subcategories Under Koszul Duality (Affine Case) -- 6.3.4 Proof of Window Theorem for DT Categories -- 6.4 Applications of Window Theorem -- 6.4.1 Derived Moduli Stacks of One Dimensional Semistable Sheaves on Surfaces -- 6.4.2 Line Bundles on Moduli Stacks -- 6.4.3 Equivalences of DT Categories for One Dimensional Stable Sheaves. , 6.4.4 Examples and Applications -- 6.5 Application to Categorical MNOP/PT Correspondence -- 6.5.1 The Moduli Stack of Semistable Objects on MNOP/PT Wall -- 6.5.2 Proof of Categorical MNOP/PT Correspondence -- 7 Categorified Hall Products on DT Categories -- 7.1 Categorified Hall Products -- 7.1.1 Derived Moduli Stacks of Extensions -- 7.1.2 Categorified Hall Algebras for Local Surfaces -- 7.1.3 Derived Moduli Stacks of Extensions of Pairs -- 7.1.4 Moduli Stacks of Extensions in AX -- 7.2 Conjectural SOD via Categorified Hall Products -- 7.2.1 Stratifications of Pnt(X, β) -- 7.2.2 Conjectures -- 7.2.3 The Formally Local Descriptions of Pnt(X, β) -- 7.3 Proofs of Conjectures 7.2.3, 7.2.4, 7.2.5, and 7.2.6 -- 7.3.1 Assumption -- 7.3.2 The Formal Local Descriptions of Pnt(X, β) Over Pnt(S, β) -- 7.3.3 The Formal Local Descriptions of the Stack of Exact Sequences -- 7.3.4 Adjoint Functors of Categorified Hall Products -- 7.3.5 Proofs of Conjectures -- 8 Some Auxiliary Results -- 8.1 Comparisons of DT Categories -- 8.1.1 Proof of Proposition 3.2.7 -- 8.1.2 Restrictions to Open Substacks -- 8.2 Compact Generation of IndCZ -- 8.2.1 Presheaves of Triangulated Categories -- 8.2.2 Compact Generation of IndCZ -- 8.3 Some Lemmas in Derived Algebraic Geometry -- 8.3.1 Equivariant Affine Derived Schemes -- 8.3.2 Proof of Proposition 3.1.3 -- 8.3.3 Proof of Proposition 3.1.5 -- 8.3.4 Proof of Lemma 6.2.15 -- 8.3.5 Proof of Lemma 6.2.17 -- 8.3.6 The Case of Formal Fibers -- 8.3.7 Proof of Lemma 6.3.6 -- 8.4 Formal Neighborhood Theorem -- 8.4.1 Formal GAGA for Good Moduli Spaces -- 8.4.2 Ext-Quivers Associated with Simple Collections -- 8.4.3 Moduli Stacks of Semistable Sheaves -- Bibliography -- Index.

Additional Edition: Print version: Toda, Yukinobu Categorical Donaldson-Thomas Theory for Local Surfaces Cham : Springer International Publishing AG,c2024 ISBN 9783031617041

Language: English