Braid group actions on derived categories of coherent sheaves

By P. Seidel, Richard P. Thomas. 2000

723 citations recorded by the source corpus.

This paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety $X$. The motivation for this is Kontsevich's homological mirror conjecture, together with the occurrence of certain braid group actions in symplectic geometry. One of the main results is that when $\dim X \geq 2$, our braid group actions are always faithful. We describe conjectural mirror symmetries between smoothings and resolutions of singularities that lead us to find examples of braid group actions arising from crepant resolutions of various singularities. Relations with the McKay correspondence and with exceptional sheaves on Fano manifolds are given. Moreover, the case of an elliptic curve is worked out in some detail.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0ad264d73f87473705374d4632e63ec1783b9543

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. B. Keller
2. Osamu Iyama
3. Emanuele Macrì
4. P. Seidel
5. T. Bridgeland
6. I. Reiten
7. Arend Bayer
8. P. Stellari

In Mathematical Physics

1. Emanuele Macrì
2. P. Stellari
3. D. Huybrechts
4. Arend Bayer
5. T. Bridgeland
6. Yukinobu Toda
7. I. Smith
8. P. Seidel

Citations in the corpus

• Silting objects, simple-minded collections, $t$-structures and co-$t$-structures for finite-dimensional algebras (2012, background): “It is straightforward to check that P1 is a 0-spherical object of Db(modΛ) in the sense of Seidel and Thomas [45].”
• Exact Results In Two-Dimensional (2,2) Supersymmetric Gauge Theories With Boundary (2013, methodology): “Construction of such autoequivalences had been given in [48, 49] and is called Seidel-Thomas twist.”
• Categorification of the braid groups (2004, background): “ngulated categories are quite widespread. They arise for instance in representation theory, for constructible sheaves on flag varieties and for coherent sheaves on Calabi-Yau varieties (cf [RouZi] and [SeiTho] for early occurrences). In this work, we suggest that not only the self-equivalences are im”
• Preprojective algebras and c‐sortable words (2010, background): “In the non-Dynkin case, Ljw is then 2-spherical in the sense of Seidel–Thomas [30].”
• Autoequivalences of derived categories on the minimal resolutions of An-singularities on surfaces (2004, background): “he definition. For an object P∈D(X×Y), an integral functor ΦP X→Y : D(X) →D(Y) 1 is defined by ΦP X→Y (−) = Rπ Y∗(P L ⊗Lπ∗ X (−)), where π X : X×Y →Xand π Y : X×Y →Y are the projections. Definition 1.2 ([ST01]). (i) We say that an object α∈D(X) is spherical if we have α⊗ω X∼=αand Homk D(X) (α,α) ∼= ( 0”
• Derived equivalences for cotangent bundles of Grassmannians via categorical 𝔰𝔩2 actions (2009, neutral): “In particular, this means that G is a spherical functor, a notion due (in varying levels of generality) to Seidel-Thomas [ ST ], Horja [Hu], Anno [A], and Rouquier [R2].”
• Dirichlet Branes and Mirror Symmetry (2009, background): “A twist functor ΦE defined by an object E ∈ D(X) is an autoequivalence of D(X) if and only if it is spherical [422]; in two dimensions, the spherical condition reads”
• Equivalences of derived catgories of sheaves on smooth stacks (2004, neutral): “e = cone{Ox0 (− 1)∨ Ox0 (− 1) → O∆X } corresponds to the twisting T ∈ Autoeq(Db(Coh(X ))) defined by T : a $→ cone{Hom(Ox0 (− 1), a)⊗Ox0 (− 1) → a} for a ∈ Db(Coh(X )) ([13]).”
• Quivers, Floer cohomology, and braid group actions (2000, background): “ion is not faithful (in the usual sense of the word) for m ≫0 [28]. Corollary 1.2 can be used to derive the faithfulness of certain braid group actions on categories which occur in algebraic geometry [38]. On the symplectic side we will study the Milnor fibres of certain singularities. The n-dimensio”
• Representations of Algebras and Related Topics (2011, background): “They yield the Seidel-Thomas [63] twist functors twSi .”
• Derived categories of coherent sheaves (2006, neutral): “9 (Seidel, Thomas [46]).”
• Cluster algebras and derived categories (2012, background): “By [133], the twist functors give rise to a (weak) action on D(Γ) of the braid group associated with Q, i.”
• Exchange graphs and Ext quivers (2011, background): “By [11] (see also [16],[21]), we know that D(ΓN Q) is a Calabi-Yau-N category, which admits a standard heart HΓ generated by simple ΓN Q-modules Se, for e ∈ Q0, each”