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Format: XI, 312 p. 159 illus. , online resource.

Edition: 1st ed. 2024.

ISBN: 9783031617058

Series Statement: Lecture Notes in Mathematics, 2350

Content: This book provides an introduction to categorical Donaldson-Thomas (DT) theory, a rapidly developing field which has close links to enumerative geometry, birational geometry, geometric representation theory and classical moduli problems in algebraic geometry. The focus is on local surfaces, i.e. the total spaces of canonical line bundles on algebraic surfaces, which form an interesting class of Calabi-Yau 3-folds. Using Koszul duality equivalences and singular support theory, dg-categories are constructed which categorify Donaldson-Thomas invariants on local surfaces. The DT invariants virtually count stable coherent sheaves on Calabi-Yau 3-folds, and play an important role in modern enumerative geometry, representation theory and mathematical physics. Requiring a basic knowledge of algebraic geometry and homological algebra, this monograph is primarily addressed to researchers working in enumerative geometry, especially Donaldson-Thomas theory, derived categories of coherent sheaves, and related areas.

Note: - Introduction -- Koszul duality equivalence -- Categorical DT theory for local surfaces -- D-critical D/K equivalence conjectures -- Categorical wall-crossing via Koszul duality -- Window theorem for DT categories -- Categori ed Hall products on DT categories -- Some auxiliary results.

In: Springer Nature eBook

Additional Edition: Printed edition: ISBN 9783031617041

Additional Edition: Printed edition: ISBN 9783031617065

Language: English

DOI: 10.1007/978-3-031-61705-8

URL: https://doi.org/10.1007/978-3-031-61705-8

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Format: xi, 309 Seiten : , Diagramme.

ISBN: 978-3-031-61704-1

Series Statement: Lecture notes in mathematics Volume 2350

Content: This book provides an introduction to categorical Donaldson-Thomas (DT) theory, a rapidly developing field which has close links to enumerative geometry, birational geometry, geometric representation theory and classical moduli problems in algebraic geometry. The focus is on local surfaces, i.e. the total spaces of canonical line bundles on algebraic surfaces, which form an interesting class of Calabi-Yau 3-folds. Using Koszul duality equivalences and singular support theory, dg-categories are constructed which categorify Donaldson-Thomas invariants on local surfaces. The DT invariants virtually count stable coherent sheaves on Calabi-Yau 3-folds, and play an important role in modern enumerative geometry, representation theory and mathematical physics. Requiring a basic knowledge of algebraic geometry and homological algebra, this monograph is primarily addressed to researchers working in enumerative geometry, especially Donaldson-Thomas theory, derived categories of coherent sheaves, and related areas

Note: Introduction -- Koszul duality equivalence -- Categorical DT theory for local surfaces -- D-critical D/K equivalence conjectures -- Categorical wall-crossing via Koszul duality -- Window theorem for DT categories -- Categori ed Hall products on DT categories -- Some auxiliary results

Additional Edition: Erscheint auch als Online-Ausgabe ISBN 978-3-031-61705-8

Language: English

Subjects: Mathematics

UID:

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