The study of derived categories is a subject that attracts increasingly many mathematicians from various fields of mathematics, including abstract algebra, algebraic geometry, representation theory, and mathematical physics. The concept of the derived category of sheaves was invented by Grothendieck and Verdier in the 1960s as a tool to express important results in algebraic geometry such as the duality theorem. In the 1970s, Beilinson, Gelfand, and Gelfand discovered that a derived category of an algebraic variety may be equivalent to that of a finite-dimensional non-commutative algebra, and Mukai found that there are non-isomorphic algebraic varieties that have equivalent derived categories. In this way, the derived category provides a new concept that has many incarnations. In the 1990s, Bondal and Orlov uncovered an unexpected parallelism between the derived categories and the birational geometry. Kontsevich's homological mirror symmetry provided further motivation for the study of derived categories. This book contains the proceedings of a conference held at the University of Tokyo in January 2011 on the current status of the research on derived categories related to algebraic geometry. Most articles are survey papers on this rapidly developing field. The book is suitable for mathematicians who want to enter this exciting field. Some basic knowledge of algebraic geometry is assumed
Author(s): Yujiro Kawamata
Series: Ems Series of Congress Reports
Publisher: European Mathematical Society
Year: 2012
Language: English
Pages: 354
Contents......Page 5
Introduction......Page 7
Introduction......Page 9
Semiorthogonal decompositions and categorical representability......Page 11
Classical representabilities and motives......Page 14
Fully faithful functors and motives......Page 17
Reconstruction of the intermediate Jacobian......Page 19
Developments and Questions......Page 22
Surfaces......Page 23
Threefolds......Page 24
Noncommutative varieties......Page 27
Higher dimensional varieties......Page 28
Other approaches......Page 29
References......Page 30
Introduction......Page 35
First properties and examples from moduli problems......Page 37
Action on (singular) cohomology......Page 38
Hochschild homology, cohomology and deformations......Page 41
The questions......Page 42
Existence of adjoints......Page 43
The algebricity assumption......Page 44
Non fully faithful functors......Page 45
Perfect complexes and good news......Page 47
Non-uniqueness of Fourier–Mukai kernels......Page 48
The remaining questions (Q3)–(Q5)......Page 50
The non-smooth case......Page 51
Some ingredients in the proof of Theorem 5.3......Page 53
Exact functors between the abelian categories of coherent sheaves......Page 56
The supported case......Page 57
More open problems......Page 60
Does full imply essentially surjective?......Page 61
Splitting functors......Page 63
Relative Fourier–Mukai functors......Page 65
References......Page 66
Flops and about: a guide by S. Cautis......Page 69
Introduction......Page 70
Cotangent bundles to Grassmannians......Page 71
Deformations of cotangent bundles......Page 72
Preliminary concepts......Page 73
Definition......Page 74
Some remarks......Page 76
Inducing equivalences......Page 77
Categorical actions on \oplus_k D(TG(k,N))......Page 78
The equivalence: an explicit description......Page 80
The inverse......Page 81
The equivalence: stratified Atiyah flops......Page 82
Equivalences and K-theory......Page 85
Geometric categorical sl_m actions......Page 86
Some remarks......Page 87
Braid group actions......Page 88
Examples......Page 89
Seidel–Thomas (spherical) twists......Page 93
P^n-twists......Page 94
Infinite twists and some geometry......Page 97
The Mukai flop......Page 100
The stratified Mukai flop of type A......Page 101
The stratified Mukai flop of type D......Page 102
Equivalences in type D......Page 105
Further topics......Page 106
References......Page 107
Introduction......Page 111
A full strong exceptional collection on X_mn......Page 112
Coherent actions and orbit categories......Page 114
References......Page 117
Introduction......Page 119
Recollection on the Segal machine......Page 120
Homology of -spaces......Page 123
Stabilization......Page 126
References......Page 129
Cluster algebras and derived categories by B. Keller......Page 131
Introduction......Page 132
First example......Page 134
Quiver mutation......Page 135
Seed mutation, cluster algebras......Page 137
Cluster algebras associated with valued quivers......Page 139
Definition......Page 142
Example: Planes in a vector space......Page 143
Example: The Grassmannian Gr(3,6)......Page 145
Example: Rectangular matrices......Page 146
Factoriality......Page 147
Parametrization of seeds by the n-regular tree......Page 149
Principal coefficients: c-vectors......Page 150
Principal coefficients: F-polynomials and g-vectors......Page 152
Tropical duality......Page 153
Product formulas for c-matrices and g-matrices......Page 155
Cluster algebras with coefficients in a semifield......Page 156
The quantum dilogarithm......Page 158
Quantum mutations and quantum cluster algebras......Page 159
Fock–Goncharov's separation formula......Page 161
The quantum separation formula......Page 162
Mutation of quivers with potential......Page 165
Ginzburg algebras......Page 167
Derived categories of dg algebras......Page 169
The derived category of the Ginzburg algebra......Page 170
Derived equivalences from mutations......Page 172
Torsion subcategories and intermediate t-structures......Page 174
Patterns of tilts and decategorification......Page 176
Reign of the tropics......Page 179
Proof of decategorification......Page 180
Proof of the quantum dilogarithm identities......Page 182
References......Page 184
Motivation and notation......Page 193
AS-regular algebras......Page 194
Fano algebras......Page 196
Generalization......Page 199
McKay correspondence......Page 200
Examples......Page 202
References......Page 203
Lagrangian-invariant sheaves and functors for abelian varieties by A. Polishchuk......Page 205
Finite Heisenberg group schemes......Page 208
1-cocycles with values in Picard stacks and twisted equivariant sheaves......Page 209
Index of a symmetric isogeny......Page 215
Kernels and functors......Page 216
Symplectic setting for abelian varieties......Page 217
Isotropic and Lagrangian pairs......Page 218
Representations associated with Lagrangian pairs and intertwining functors......Page 222
Invariants of a generalized Lagrangian pair......Page 224
Lagrangian correspondences for ess-abelian varieties......Page 237
LI-kernels and functors......Page 240
Central extensions related to LI-endofunctors......Page 252
References......Page 257
Introduction......Page 259
Preliminaries and examples......Page 262
Comparison with a Bezrukavnikov–Kashiwara perverse t-structure......Page 264
Hypercohomology vanishing characterization of Per(Y) and of GV_m (X) with m \leq 0......Page 269
Commutative algebra filtration on Per(Y), describing GV_m (X) with m >0......Page 271
Geometric applications......Page 273
Generic vanishing theorems......Page 274
Vanishing of higher direct images......Page 277
Bounding the holomorphic Euler characteristic and applications to irregular varieties......Page 278
Moduli spaces of vector bundles......Page 280
Appendix: some homological commutative algebra......Page 282
References......Page 284
Introduction......Page 287
A related problem......Page 288
The example......Page 289
Proof of the theorem......Page 291
References......Page 293
Introduction......Page 295
(Semi)stable sheaves......Page 296
Moduli theory of stable sheaves......Page 297
Perfect obstruction theory......Page 298
Virtual class......Page 301
Behrend function......Page 303
Rank one DT invariants......Page 305
GW/DT correspondence......Page 307
Stable pair theory......Page 309
DT/PT, rationality......Page 311
Flop formula......Page 314
Stable pairs on local K3 surfaces......Page 315
Multiple cover formula......Page 318
Computation of J(r, beta , n)......Page 320
Construction of DT type invariants w.r.t. Bridgeland stability......Page 321
Construction of Bridgeland stability conditions on 3-folds......Page 322
References......Page 324
Introduction......Page 327
Base extension......Page 329
Completion of noetherian abelian categories......Page 330
Functors......Page 336
Formal flatness......Page 338
Ext-groups......Page 340
Ampleness......Page 345
Lifting and base change......Page 349
References......Page 351
List of Contributors......Page 353
