Derived categories of coherent sheaves

By T. Bridgeland. 2006

266 citations recorded by the source corpus.

We discuss derived categories of coherent sheaves on algebraic varieties. We focus on the case of non-singular Calabi-Yau varieties and consider two unsolved problems: proving that birational varieties have equivalent derived categories, and computing the group of derived autoequivalences. We also introduce the space of stability conditions on a triangulated category and explain its relevance to these two problems.

Subfields: Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/7f7bf7efb969ffd81d4ab701228406cd192bb680

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Citations in the corpus

Duality in two-dimensional (2,2) supersymmetric non-Abelian gauge theories (2011, methodology): “5 is related to the following equivalences, found by BondalOrlov [19, 14, 20]: For even N D(QS) ∼= 〈CS,O1, .”
Hirzebruch–Riemann–Roch‐type formula for DG algebras (2007, background): “Orlov’s research on the derived categories of smooth schemes [9].”
Lefschetz decompositions and categorical resolutions of singularities (2006, background): “Unfortunately, [7] seems to be too restrictive, while [20] and [12] are not sufficiently worked out.”
Moduli stacks and invariants of semistable objects on K3 surfaces (2007, background): “As stated in [8], [17], it is interesting to compare the formula (4) with the work of Borcherds [3] on the product expansions of the automorphic forms.”
Homological projective duality (2005, background): “1 we can deduce that there is an equivalence of derived categories between the following two varieties related by a special birational transformation called a flop (it is conjectured in [BO2] that the derived categories of any pair of algebraic varieties related by a flop are equivalent).”
Semiorthogonal decompositions in algebraic geometry (2014, methodology): “This paper can be considered as a continuation and a development of the ICM 2002 talk [7] of Alexei Bondal and Dmitri Orlov.”
A Guided Tour Through the Garden of Noncommutative Motives (2011, background): “Since then, Drinfeld, Kaledin, Kontsevich, Orlov, Van den Bergh, and others, have made important advances; see [9, 10, 16, 17, 25, 30, 31, 32, 33].”
Hyperplane sections and derived categories (2005, background): “Finally, Bondal and Orlov [BO2, BO4] stated a theorem describing the structure of the derived category of a complete intersection of any number of quadrics.”
The derived category of a GIT quotient (2012, background): “In Section 4, we apply Theorem 1.1 to construct new examples of derived equivalences and embeddings resulting from birational transformations, as conjectured by Bondal & Orlov [7].”