*-Quantizations of Fourier–Mukai Transforms

Arinkin, Dima; Block, Jonathan; Pantev, Tony

doi:10.1007/s00039-013-0233-8

Cited by 3 publications

(6 citation statements)

References 26 publications

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“…Recall that the signature of L is (21,2). Choose a 21-dimensional positive definite real subspace V ⊂ L ⊗ R such that V ∩ L = h 2 , Q, T .…”

Section: Non-emptinessmentioning

confidence: 99%

“…. 5 There is also related work [2] using deformation quantisation. 6 Alternatively use the right adjoint ΦR (6.6), which is a left inverse to the fully faithful ΦP .…”

Section: (Hochschild) (Co)homologymentioning

confidence: 99%

“…Arguing as in the proof of Theorem 7.7, its composition with its right adjoint gives a kernel whose restriction to Shrinking Z further if necessary, the resulting full and faithful embeddings D(S t ) → D(X t ) have image in A Xt because this is true at t = 0 and being orthogonal to the pullbacks from X t of O Xt , O Xt (1), O Xt (2) is also an open condition. Since the Serre functor on both D(S t ) and A Xt is the shift by [2], and since D(S t ) is non-zero and A Xt is indecomposable, any fully faithful functor D(S t ) → A Xt is necessarily an equivalence [21, Prop. 1.54].…”

Section: Up To Units In C[[t]]mentioning

confidence: 99%

“…This looks like (a Tate twist of) H 2 of a K3 surface S, but the intersection forms have different signatures: (20,2) for H 4 prim (X, Z) versus (19, 3) for H 2 (S, Z)(−1). However, one can often find a codimension-1 sub-Hodge structure of signature (19,2) common to both. For the K3 surface it is H 2 prim (S, Z)(−1), the orthogonal to some primitive ample class .…”

Section: Introductionmentioning

confidence: 99%

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Hodge theory and derived categories of cubic fourfolds

Addington¹,

Thomas²

2014

Duke Math. J.

119

384

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Abstract. Cubic fourfolds behave in many ways like K3 surfaces. Certain cubics -conjecturally, the ones that are rational -have specific K3 surfaces associated to them geometrically. Hassett has studied cubics with K3 surfaces associated to them at the level of Hodge theory, and Kuznetsov has studied cubics with K3 surfaces associated to them at the level of derived categories.These two notions of having an associated K3 surface should coincide. We prove that they coincide generically: Hassett's cubics form a countable union of irreducible Noether-Lefschetz divisors in moduli space, and we show that Kuznetsov's cubics are a dense subset of these, forming a non-empty, Zariski open subset in each divisor.

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“…Recall that the signature of L is (21,2). Choose a 21-dimensional positive definite real subspace V ⊂ L ⊗ R such that V ∩ L = h 2 , Q, T .…”

Section: Non-emptinessmentioning

confidence: 99%

“…. 5 There is also related work [2] using deformation quantisation. 6 Alternatively use the right adjoint ΦR (6.6), which is a left inverse to the fully faithful ΦP .…”

Section: (Hochschild) (Co)homologymentioning

confidence: 99%

Section: Up To Units In C[[t]]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Hodge theory and derived categories of cubic fourfolds

Addington¹,

Thomas²

2014

Duke Math. J.

119

384

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“…We are partially guided in this study by the analogy with modules over algebraic (or complex analytic) deformation quantization algebras that have been studied e.g. in [19], [5], [2], [4], [26]. In the case when O N C X is obtained from a torsion-free connection on X we mimic our construction of the dg-resolution for modules.…”

mentioning

confidence: 99%

DG-resolutions of NC-smooth thickenings and NC-Fourier–Mukai transforms

Polishchuk

2014

Math. Ann.

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Abstract. We give a construction of NC-smooth thickenings (a notion defined by Kapranov [17]) of a smooth variety equipped with a torsion free connection. We show that a twisted version of this construction realizes all NC-smooth thickenings as 0th cohomology of a differential graded sheaf of algebras, similarly to Fedosov's construction in [12]. We use this dg resolution to construct and study sheaves on NC-smooth thickenings. In particular, we construct an NC version of the Fourier-Mukai transform from coherent sheaves on a (commutative) curve to perfect complexes on the canonical NC-smooth thickening of its Jacobian. We also define and study analytic NC-manifolds. We prove NC-versions of some of GAGA theorems, and give a C ∞ -construction of analytic NC-thickenings that can be used in particular for Kähler manifolds with constant holomorphic sectional curvature. Finally, we describe an analytic NC-thickening of the Poincaré line bundle for the Jacobian of a curve, and the corresponding Fourier-Mukai functor, in terms of A ∞ -structures.

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