On the derived category of coherent sheaves on a 5-dimensional Fano variety

2005 ◽  
Vol 340 (12) ◽  
pp. 889-893 ◽  
Author(s):  
Alexander Samokhin
Keyword(s):  
Derived Category ◽  
Fano Variety ◽  
Coherent Sheaves

scholarly journals A note on thick subcategories of stable derived categories

2013 ◽  
Vol 212 ◽  
pp. 87-96
Author(s):  
Henning Krause ◽  
Greg Stevenson
Keyword(s):  
Derived Categories ◽  
Derived Category ◽  
Complete Intersections ◽  
Line Bundles ◽  
Finitely Generated ◽  
Exact Category ◽  
Coherent Sheaves ◽  
Noetherian Scheme ◽  
Finitely Generated Modules ◽  
Local Complete Intersections

AbstractFor an exact category having enough projective objects, we establish a bijection between thick subcategories containing the projective objects and thick subcategories of the stable derived category. Using this bijection, we classify thick subcategories of finitely generated modules over strict local complete intersections and produce generators for the category of coherent sheaves on a separated Noetherian scheme with an ample family of line bundles.

scholarly journals Completing perfect complexes

2020 ◽  
Vol 296 (3-4) ◽  
pp. 1387-1427 ◽  
Author(s):  
Henning Krause
Keyword(s):  
Derived Categories ◽  
Derived Category ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Direct Construction ◽  
Coherent Sheaves ◽  
Perfect Complexes ◽  
Coherent Ring ◽  
Abelian Categories ◽  
Finitely Presented

Abstract This note proposes a new method to complete a triangulated category, which is based on the notion of a Cauchy sequence. We apply this to categories of perfect complexes. It is shown that the bounded derived category of finitely presented modules over a right coherent ring is the completion of the category of perfect complexes. The result extends to non-affine noetherian schemes and gives rise to a direct construction of the singularity category. The parallel theory of completion for abelian categories is compatible with the completion of derived categories. There are three appendices. The first one by Tobias Barthel discusses the completion of perfect complexes for ring spectra. The second one by Tobias Barthel and Henning Krause refines for a separated noetherian scheme the description of the bounded derived category of coherent sheaves as a completion. The final appendix by Bernhard Keller introduces the concept of a morphic enhancement for triangulated categories and provides a foundation for completing a triangulated category.

Fourier-Mukai Transforms in Algebraic Geometry

2007 ◽  
Author(s):  
D. Huybrechts
Keyword(s):  
Algebraic Geometry ◽  
Smooth Projective Variety ◽  
Derived Category ◽  
Point Of View ◽  
Basic Knowledge ◽  
Research Directions ◽  
Coherent Sheaves ◽  
Geometric Point ◽  
Singular Cohomology

This book provides a systematic exposition of the theory of Fourier-Mukai transforms from an algebro-geometric point of view. Assuming a basic knowledge of algebraic geometry, the key aspect of this book is the derived category of coherent sheaves on a smooth projective variety. The derived category is a subtle invariant of the isomorphism type of a variety, and its group of autoequivalences often shows a rich structure. As it turns out — and this feature is pursued throughout the book — the behaviour of the derived category is determined by the geometric properties of the canonical bundle of the variety. Including notions from other areas, e.g., singular cohomology, Hodge theory, abelian varieties, K3 surfaces; full proofs and exercises are provided. The final chapter summarizes recent research directions, such as connections to orbifolds and the representation theory of finite groups via the McKay correspondence, stability conditions on triangulated categories, and the notion of the derived category of sheaves twisted by a gerbe.

The wall-crossing behavior for Bridgeland’s stability conditions on abelian and K3 surfaces

2018 ◽  
Vol 2018 (735) ◽  
pp. 1-107 ◽  
Author(s):  
Hiroki Minamide ◽  
Shintarou Yanagida ◽  
Kōta Yoshioka
Keyword(s):  
Stability Condition ◽  
Derived Category ◽  
The Other ◽  
Abelian Surface ◽  
Stability Conditions ◽  
Abelian Surfaces ◽  
Coherent Sheaves ◽  
Fundamental Properties ◽  
Wall Crossing ◽  
Wall Crossing Formula

AbstractThe wall-crossing behavior for Bridgeland’s stability conditions on the derived category of coherent sheaves on K3 or abelian surface is studied. We introduce two types of walls. One is called the wall for categories, where thet-structure encoded by stability condition is changed. The other is the wall for stabilities, where stable objects with prescribed Mukai vector may get destabilized. Some fundamental properties of walls and chambers are studied, including the behavior under Fourier–Mukai transforms. A wall-crossing formula of the counting of stable objects will also be derived. As an application, we will explain previous results on the birational maps induced by Fourier–Mukai transforms on abelian surfaces. These transformations turns out to coincide with crossing walls of certain property.

scholarly journals Base change for semiorthogonal decompositions

2011 ◽  
Vol 147 (3) ◽  
pp. 852-876 ◽  
Author(s):  
Alexander Kuznetsov
Keyword(s):  
Algebraic Variety ◽  
Base Change ◽  
Derived Category ◽  
Intermediate Step ◽  
Coherent Sheaves ◽  
Base Scheme ◽  
Compatible System ◽  
Perfect Complexes

AbstractLet X be an algebraic variety over a base scheme S and ϕ:T→S a base change. Given an admissible subcategory 𝒜 in 𝒟b(X), the bounded derived category of coherent sheaves on X, we construct under some technical conditions an admissible subcategory 𝒜T in 𝒟b(X×ST), called the base change of 𝒜, in such a way that the following base change theorem holds: if a semiorthogonal decomposition of 𝒟b (X) is given, then the base changes of its components form a semiorthogonal decomposition of 𝒟b (X×ST) . As an intermediate step, we construct a compatible system of semiorthogonal decompositions of the unbounded derived category of quasicoherent sheaves on X and of the category of perfect complexes on X. As an application, we prove that the projection functors of a semiorthogonal decomposition are kernel functors.

scholarly journals Residual categories for (co)adjoint Grassmannians in classical types

2021 ◽  
Vol 157 (6) ◽  
pp. 1172-1206
Author(s):  
Alexander Kuznetsov ◽  
Maxim Smirnov
Keyword(s):  
Automorphism Group ◽  
Cohomology Ring ◽  
Quantum Cohomology ◽  
Algebraic Groups ◽  
Fano Variety ◽  
Flag Varieties ◽  
Picard Number ◽  
Coherent Sheaves ◽  
Small Quantum ◽  
Exceptional Collection

In our previous paper we suggested a conjecture relating the structure of the small quantum cohomology ring of a smooth Fano variety of Picard number 1 to the structure of its derived category of coherent sheaves. Here we generalize this conjecture, make it more precise, and support it by the examples of (co)adjoint homogeneous varieties of simple algebraic groups of Dynkin types $\mathrm {A}_n$ and $\mathrm {D}_n$ , that is, flag varieties $\operatorname {Fl}(1,n;n+1)$ and isotropic orthogonal Grassmannians $\operatorname {OG}(2,2n)$ ; in particular, we construct on each of those an exceptional collection invariant with respect to the entire automorphism group. For $\operatorname {OG}(2,2n)$ this is the first exceptional collection proved to be full.

Derived Category and Canonical Bundle – I

2007 ◽  
Author(s):  
D. Huybrechts
Keyword(s):  
Elliptic Curves ◽  
Derived Categories ◽  
Derived Category ◽  
Canonical Bundle ◽  
Smooth Curves ◽  
Coherent Sheaves

This chapter is devoted to results by Bondal and Orlov which show that for varieties with ample (anti-)canonical bundle, the bounded derived category of coherent sheaves determines the variety. Except for the case of elliptic curves, this settles completely the classification of derived categories of smooth curves. The complexity of the derived category is reflected by its group of autoequivalences. This is studied by means of ample sequences.

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