Relative Calabi-Yau structures II: Shifted Lagrangians in the moduli of objects

Brav, Christopher; Dyckerhoff, Tobias

doi:10.48550/arxiv.1812.11913

Cited by 5 publications

(12 citation statements)

References 13 publications

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“…By Proposition 3.4 in [TV07] and Example 3.7 in [BD19], the moduli space M C has the following universal property:…”

Section: Definition 3 ([Bd19]) a Non-commutative Calabi-yau Of Dimens...mentioning

confidence: 99%

“…In particular we work with moduli of pseudo-perfect objects in a Calabi-Yau 3-category. We recall the relevant definition given in [BD19].…”

Section: Definition 2 ([Tv07]) Define a Simplicial Presheaf: Mmentioning

confidence: 99%

“…Here, a Calabi-Yau structure of dimension d is a negative cyclic chain θ : k[d] → HC − (C) satisfying a certain non-degeneracy condition, see [BD19]. Given such a non-commutative Calabi-Yau C, the moduli of pseudo-perfect objects in C is defined in the following definition.…”

Section: Definition 3 ([Bd19]) a Non-commutative Calabi-yau Of Dimens...mentioning

confidence: 99%

“…The objects are all admissible functors from the category of derived affine schemes to the ∞-category of spaces. Definition 4 (Example 3.7, [BD19]). The moduli space of objects M C in a compactly generated dgcategory C is the prestack given on every affine U by…”

Section: Definition 3 ([Bd19]) a Non-commutative Calabi-yau Of Dimens...mentioning

confidence: 99%

“…This follows from the fact that the derived moduli spaces of coherent sheaves/perfect complexes on a Calabi-Yau 3-fold has a −1-shifted symplectic structure. See [PTVV13] for the case of compact Calabi-Yau 3-folds, and [BD19] for noncompact Calabi-Yau 3-folds. Then [BBJ], [BBBBJ15] show that there is a d-critical locus structure obtained by truncating the −1-shifted symplectic structure from derived geometry.…”

Section: Introductionmentioning

confidence: 99%

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D-critical loci for local toric Calabi-Yau 3-folds

Katz¹,

Shi²

2021

Preprint

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The notion of a d-critical locus is an ingredient in the definition of motivic Donaldson-Thomas invariants by [BJM19]. There is a canonical d-critical locus structure on the Hilbert scheme of dimension zero subschemes on local toric Calabi-Yau 3-folds. This is obtained by truncating the −1-shifted symplectic structure on the derived moduli stack [BBBBJ15]. In this paper we show the canonical d-critical locus structure has critical charts consistent with the description of Hilbert scheme as a degeneracy locus [BBS13]. In particular, the canonical d-critical locus structure is isomorphic to the one constructed in [KS21] for local P 2 and local F n .

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“…By Proposition 3.4 in [TV07] and Example 3.7 in [BD19], the moduli space M C has the following universal property:…”

Section: Definition 3 ([Bd19]) a Non-commutative Calabi-yau Of Dimens...mentioning

confidence: 99%

“…In particular we work with moduli of pseudo-perfect objects in a Calabi-Yau 3-category. We recall the relevant definition given in [BD19].…”

Section: Definition 2 ([Tv07]) Define a Simplicial Presheaf: Mmentioning

confidence: 99%

Section: Definition 3 ([Bd19]) a Non-commutative Calabi-yau Of Dimens...mentioning

confidence: 99%

Section: Definition 3 ([Bd19]) a Non-commutative Calabi-yau Of Dimens...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

D-critical loci for local toric Calabi-Yau 3-folds

Katz¹,

Shi²

2021

Preprint

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Dimensional reduction in cohomological Donaldson-Thomas theory

Kinjo

2021

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For oriented −1-shifted symplectic derived Artin stacks, Ben-Bassat-Brav-Bussi-Joyce introduced certain perverse sheaves on them which can be regarded as sheaf theoretic categorifications of the Donaldson-Thomas invariants. In this paper, we prove that the hypercohomology of the above perverse sheaf on the −1-shifted cotangent stack over a quasi-smooth derived Artin stack is isomorphic to the Borel-Moore homology of the base stack up to a certain shift of degree. This is a global version of the dimensional reduction theorem due to Davison.We give two applications of our main theorem. Firstly, we apply it to the study of the cohomological Donaldson-Thomas invariants for local surfaces. Secondly, regarding our main theorem as a version of Thom isomorphism theorem for dual obstruction cones, we propose a sheaf theoretic construction of the virtual fundamental classes for quasi-smooth derived Artin stacks. Contents 1. Introduction 1 2. Shifted symplectic structures and vanishing cycles 5 3. Dimensional reduction for schemes 16 4. Dimensional reduction for stacks 21 5. Applications 36 Appendix A. Remarks on the determinant functor 39 References 44. By taking the derived functor of f * we obtain Rf * . When f is a smooth morphism, f * is nothing but the restriction functor. In general f * is constructed by taking simplicial covers, but we use pullback functors only for smooth morphisms in this paper, so we do not need this general construction. To define Rf ! and f ! we use the Verdier duality functor. Arguing as in [LO08, §3], we can construct the dualizing complex ω X ∈ D (b) c (X lis−an , Q) and define the Verdier duality functor D X := RHom(−, ω X ) : D (−) c (X lis−an , Q) op → D (+) c (X lis−an , Q).

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Purity and 2-Calabi-Yau categories

Davison¹

2021

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For various 2-Calabi-Yau categories C for which the stack of objects M has a good moduli space p : M → M, we establish purity of the mixed Hodge module complex p ! Q M . We do this by using formality in 2CY categories, along with étale neighbourhood theorems for stacks, to prove that the morphism p is modelled étale-locally by the semisimplification morphism from the stack of modules of a preprojective algebra. Via the integrality theorem in cohomological Donaldson-Thomas theory we prove purity of p ! Q M . It follows that the Beilinson-Bernstein-Deligne-Gabber decomposition theorem for the constant sheaf holds for the morphism p, despite the possibly singular and stacky nature of M. We use this to define cuspidal cohomology for M, which is conjecturally a complete space of generators for the BPS algebra associated to C .We prove purity of the Borel-Moore homology of the moduli stack M, provided its good moduli space M is projective, or admits a suitable contracting C * -action. In particular, when M is the moduli stack of Gieseker-semistable sheaves on a K3 surface, this proves a conjecture of Halpern-Leistner. We use these results to moreover prove purity for several stacks of coherent sheaves that do not admit a good moduli space.Without the usual assumption that r and d are coprime, we prove that the Borel-Moore homology of the stack of semistable degree d rank r Higgs sheaves is pure and carries a perverse filtration with respect to the Hitchin base, generalising the usual perverse filtration for the Hitchin system to the case of singular stacks of Higgs sheaves.

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Relative Calabi-Yau structures II: Shifted Lagrangians in the moduli of objects

Cited by 5 publications

References 13 publications

D-critical loci for local toric Calabi-Yau 3-folds

D-critical loci for local toric Calabi-Yau 3-folds

Dimensional reduction in cohomological Donaldson-Thomas theory

Purity and 2-Calabi-Yau categories

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