The derived category of a GIT quotient

By Daniel Halpern-Leistner. 2012

184 citations recorded by the source corpus.

Given a quasiprojective algebraic variety with a reductive group action, we describe a relationship between its equivariant derived category and the derived category of its geometric invariant theory quotient. This generalizes classical descriptions of the category of coherent sheaves on projective space and categorifies several results in the theory of Hamiltonian group actions on projective manifolds. This perspective generalizes and provides new insight into examples of derived equivalences between birational varieties. We provide a criterion under which two different GIT quotients are derived equivalent, and apply it to prove that any two generic GIT quotients of an equivariantly Calabi-Yau projective-over-affine manifold by a torus are derived equivalent.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f1c870a40cda02dafa3229936ac61fe1be952fa1

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. Andrei Neguct
2. E. Gorsky
3. Jacob Rasmussen
4. Will Donovan
5. A. Okounkov
6. E. Segal
7. Steven V. Sam
8. Daniel Halpern-Leistner

In Mathematical Physics

1. Andrei Neguct
2. E. Gorsky
3. Jacob Rasmussen
4. Will Donovan
5. A. Okounkov
6. E. Segal
7. Steven V. Sam
8. Daniel Halpern-Leistner

Citations in the corpus

• Combinatorial constructions of derived equivalences (2016, background): “ define Lλ>0 to be the projection of this class onto the subspace spanned by weights which pair positively with λ. For any cocharacter λ we define ηλ := hλ,Lλ>0i. This agrees with the η defined in [HL]. We note also that ηλ = ηwλ for any w ∈ W, and ηλ = η−λ because X is quasi-symmetric. Definition”
• Flag Hilbert schemes, colored projectors and Khovanov-Rozansky homology (2016, background): “Similar decompositions of equivariant derived categories with respect to Białynicki-Birula strata were studied in [35], and we plan to study the relation between the categorical diagonalization framework and [35] in the future work.”
• Autoequivalences of derived categories via geometric invariant theory (2013, unknown): “If ωX |Z has weight 0 with respect to λ±, then η+ = η− (see [13]).”
• Non-commutative resolutions of quotient singularities for reductive groups (2015, unknown): “Other papers which discuss the homological properties of categories of Gequivariant modules and more generally coherent sheaves are [4,25].”
• Lectures on K-theoretic computations in enumerative geometry (2015, background): “This is a special case of the theory developed in [39], where many applications of these techniques can also be found.”
• Window shifts, flop equivalences and Grassmannian twists (2012, methodology): “Recent work of Halpern-Leistner [Hal12] and Ballard, Favero and Katzarkov [BFK12], also using ideas from [HHP08, Seg11], develops a general theory of derived equivalences corresponding to certain variations of GIT.”