D-Equivalence and K-Equivalence

By Y. Kawamata. 2002

255 citations recorded by the source corpus.

Let X and Y be smooth projective varieties over C. They are called Dequivalent if their derived categories of bounded complexes of coherent sheaves are equivalent as triangulated categories, and K-equivalent if they are birationally equivalent and the pull-backs of their canonical divisors to a common resolution coincide. We expect that the two equivalences coincide for birationally equivalent varieties. We shall provide a partial answer to the above problem in this paper.

Subfields: Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/d0b8c0d87de92e70246999a857a093bca492e1b7

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. Yukinobu Toda
2. T. Bridgeland
3. Sabin Cautis
4. Emanuele Macrì
5. P. Aspinwall
6. G. Moore
7. Arend Bayer
8. Joel Kamnitzer

In Statistical and Nonlinear Physics

1. Sabin Cautis
2. T. Bridgeland
3. Yukinobu Toda
4. Joel Kamnitzer
5. G. Moore
6. Y. Kawamata
7. P. Aspinwall
8. Balázs Szendrői

Citations in the corpus

• Derived categories of sheaves: a skimming (2005, unknown): “Their conjecture was proven by Bridgeland [6] and later generalized by Chen [10] and Kawamata [18].”
• Autoequivalences of derived categories via geometric invariant theory (2013, unknown): “Then according to a theorem of Kawamata [18], the functor (prX)∗(pr ′ X) ∗ : D(X ′) → D(X) is an exact equivalence.”
• Orbifolds and Stringy Topology (2007, background): “Indeed, work on the McKay correspondence in algebraic geometry is usually phrased in such terms; see Bridgeland, King, and Reid [32] and the papers of Kawamata [81–83] for more on this influential approach.”
• Autoequivalences of derived categories on the minimal resolutions of An-singularities on surfaces (2004, background): “Given a smooth projective variety X, it is an interesting problem to find all the varieties Y with D(X) ∼ D(Y ). In dimension 2, the answer is given by Bridgeland and Maciocia in [BM01], and Kawamata [ Kaw02 ] and in dimension 3 some results are shown by Toda [Tod03].”
• Knot homology via derived categories of coherent sheaves II, $\mathfrak{sl}_{m}$ case (2007, methodology): “In the simplest situation, the two projections from Y 1,m−2,1 to Y 1,m−1 and Ym−1,1 describe a Mukai flop. The fact that OZ1 1,m−1 induces an equivalence D(Y 1,m −) →D(Ym , ) was proven by Kawamata in [Kaw] and Namikawa in [Na]. More generally, if βcontains a string (1,m−1) starting in the ith positi”
• Log Crepant Birational Maps and Derived Categories (2003, background): “Then the proofs of the statements (1) through (3) are similar to those in [5].”
• Dirichlet Branes and Mirror Symmetry (2009, background): “This conjecture was made by Kawamata [300], after Bridgeland [61] had already settled the three-dimensional case, where Conjecture 4.”
• Derived categories of coherent sheaves (2006, background): “A more direct proof has since been found by Kawamata [26] using results of Van den Bergh [50].”
• Equivalences of derived catgories of sheaves on smooth stacks (2004, methodology): “Section 7 is devoted to applications similar to those appearing in [9].”
• McKay correspondence for elliptic genera (2002, background): “It is conjectured in [20] that K-equivalent varieties have equivalent derived categories.”
• Deformations and Fourier-Mukai transforms (2005, background): “Kawamata [10] conjectured that derived categories are equivalent under K-equivalence.”