Derived Equivalences by Quantization

By D. Kaledin. 2005

72 citations recorded by the source corpus.

We assume given a smooth symplectic (in the algebraic sense) resolution $X$ of an affine algebraic variety $Y$, and we prove that, possibly after replacing $Y$ with an etale neighborhood of a point, the derived category of coherent sheaves on $X$ is equivalent to the dervied category of finitely generated left modules over a non-commutative algebra $R$, a non-commutative resolution of $Y$ in a sense close to that of M. Van den Bergh. We also prove some applications, such as: two resolutions are derived-equivalent; every resolution $X$ admits a "resolution of the diagonal"; the cohomology groups of the fibers of the map $X \to Y$ are spanned by fundamental classes of algebraic cycles.

Subfields: Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/620a81a3fe4a822a5a8250e30668c00d02867613

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. Daniel Halpern-Leistner
2. Alexander Kuznetsov
3. Graham J. Leuschke
4. Michel van den Bergh
5. Y. Kawamata
6. R. Buchweitz
7. Steven V. Sam
8. T. Bridgeland

In Statistical and Nonlinear Physics

1. Daniel Halpern-Leistner
2. Michel van den Bergh
3. Alexander Kuznetsov
4. Steven V. Sam
5. Y. Kawamata
6. Graham J. Leuschke
7. Emanuele Macrì
8. P. Stellari

Citations in the corpus

• Lefschetz decompositions and categorical resolutions of singularities (2006, result): “Another work in this area was done by Kaledin [ 11 ].”
• Combinatorial constructions of derived equivalences (2016, methodology): “.2. Then for any pair of ℓ,ℓ′ ∈ Pic(BG) R such that Xss 0 = Xs 0 , we have a derived equivalence Db(Xss 0 (ℓ)/G) → Db(Xss 0 (ℓ ′)/G). The proof is somewhat more geometric and explicit than the one in [Kal], as it avoids deformation quantization and reduction to positive characteristic. It also makes”
• Non-commutative desingularization of determinantal varieties I (2009, background): “s defined in [21]. Non-commutative desingularizations occurred probably first in theoretical physics (e.g. [2]) but they have recently been encountered in a number of purely mathematical contexts (e.g. [3, 14, 15, 17, 20]). The next result is a description by generators and relations of the non-commut”
• Elliptic stable envelopes (2016, unknown): “[7–10, 43].”
• Derived categories of toric varieties III (2006, background): “The derived McKay correspondence was already proved for finite subgroups of SL(2,C), SL(3,C) (Bridgeland-King-Reid [6]), and Sp(2n,C) (Bezrukavnikov-Kaledin [2]).”