Orthogonally spherical objects and spherical fibrations

Anno, Rina; Logvinenko, Timothy

doi:10.1016/j.aim.2015.08.027

Cited by 8 publications

(27 citation statements)

References 30 publications

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“…Dually, we find that another window shift autoequivalence, namely ω −1,0 , can be described in terms of a cotwist [AL10] around a spherical functor with source D b (X + ): we defer a precise statement until Section 3.2 (Theorem 3.13). As a pleasing corollary, we find that a twist and a cotwist on D b (X + ) are related (Corollary 3.14).…”

Section: Introductionmentioning

confidence: 89%

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Window shifts, flop equivalences and Grassmannian twists

Donovan

Segal

2014

Compositio Math.

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16.04.15 KB. OK to add accepted version to spiralWe introduce a new class of autoequivalences that act on the derived categories of certain vector bundles over Grassmannians. These autoequivalences arise from Grassmannian flops: they generalize Seidel-Thomas spherical twists, which can be seen as arising from standard flops. We first give a simple algebraic construction, which is well-suited to explicit computations. We then give a geometric construction using spherical functors which we prove is equivalent

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Section: Introductionmentioning

confidence: 89%

“…. General theory [AL10] says that T F is an equivalence iff C F is an equivalence, given the fact that X + and Y + are Calabi-Yau.…”

Section: Consequently We Havementioning

confidence: 99%

Window shifts, flop equivalences and Grassmannian twists

Donovan

Segal

2014

Compositio Math.

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“…The same is true if A and B are admissible subcategories of these, because the projection functors are induced by kernels [25]. 3 It is also possible to do business with derived 2 Rouquier requires the triangle (1.2) to be split, but we do not. Both he and Anno require a certain natural map R → CL to be an isomorphism, but this is difficult to check in practice, and in our proof of Theorem 1 below we will see that any isomorphism R ∼ = CL will do.…”

Section: 1 Definitionmentioning

confidence: 97%

New derived symmetries of some hyperkähler varieties

Addington¹

2016

Alg. Geom.

176

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We construct a new autoequivalence of the derived category of the Hilbert scheme of n points on a K3 surface, and of the variety of lines on a smooth cubic 4-fold. For Hilb 2 and the variety of lines, we use the theory of spherical functors; to deal with Hilb n for n > 2, we develop a theory of P-functors. We conjecture that the same construction yields an autoequivalence for any moduli space of sheaves on a K3 surface. In an appendix we give a cohomology and base change criterion which is well known to experts, but not well documented.

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“…Part (1) is the key to establishing that the J-twist is an autoequivalence, and part (2) shows the link to spherical functors (see [AL1,AL2]). Both parts follow quite easily from the fact that R is a hypersurface singularity, using matrix factorisations.…”

Section: Amentioning

confidence: 99%

Twists and braids for general 3-fold flops

Donovan

Wemyss

2019

J. Eur. Math. Soc.

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Given a quasi-projective 3-fold X with only Gorenstein terminal singularities, we prove that the flop functors beginning at X satisfy higher degree braid relations, with the combinatorics controlled by a real hyperplane arrangement H. This leads to a general theory, incorporating known special cases with degree 3 braid relations, in which we show that higher degree relations can occur even for two smooth rational curves meeting at a point. This theory yields an action of the fundamental group of the complexified complement π 1 (C n \H C ) on the derived category of X, for any such 3-fold that admits individually floppable curves. We also construct such an action in the more general case where individual curves may flop analytically, but not algebraically, and furthermore we lift the action to a form of affine pure braid group under the additional assumption that X is Q-factorial.Along the way, we produce two new types of derived autoequivalences. One uses commutative deformations of the scheme-theoretic fibre of a flopping contraction, and the other uses noncommutative deformations of the fibre with reduced scheme structure, generalising constructions of Toda and the authors [T07, DW1] which considered only the case when the flopping locus is irreducible. For type A flops of irreducible curves, we show that the two autoequivalences are related, but that in other cases they are very different, with the noncommutative twist being linked to birational geometry via the Bridgeland-Chen [B02, C02] flop-flop functor. J ) → 0. Definition 2.18. With notation as above, in particular A := End R (N ), we define the left mutation of N at N J asand set ν J A := End R (ν J N ).One of the key properties of mutation is that it always gives rise to a derived equivalence. With the setup as above, the derived equivalence between A and ν J A is given by a tilting A-module T J constructed as follows. There is an exact sequenceobtained by dualizing (2.K). Applying Hom R (N, −) induces (b * J ·) : Hom R (N, N J ) → Hom R (N, U J ), so denoting the cokernel by C J there is an exact sequenceCorollary 5.12. With the global quasi-projective flops setup of 2.2, and notation in (5.I), for all y ∈ D(QcohX), there is a functorial triangleProof. By the above, all kernels in (5.K) have right adjoints given by applying (−) R := RHomX ×X (−, π ! 2 OX ). We know that D R J gives JTwist p , and clearly the right adjoint of the identity functor is the identity. Further by 5.9 the FM functor given by Q isThe Projective Twist is an Equivalence. A formal consequence of the twist definition 5.4 is the following intertwinement lemma.Proposition 5.13. The following diagram is naturally commutative.

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Orthogonally spherical objects and spherical fibrations

Cited by 8 publications

References 30 publications

Window shifts, flop equivalences and Grassmannian twists

Window shifts, flop equivalences and Grassmannian twists

New derived symmetries of some hyperkähler varieties

Twists and braids for general 3-fold flops

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