Moduli stacks of semistable sheaves and representations of Ext-quivers

Toda, Yukinobu

doi:10.48550/arxiv.1710.01841

Cited by 3 publications

(4 citation statements)

References 19 publications

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“…Our results have several other immediate implications. We can recover the main theorem of [AS18, Theorem 1.1] on symplectic resolutions of moduli spaces M X,H (v) of H-semistable sheaves on K3 surfaces via variations of GIT quotients of quiver varieties, by combining Theorem 1.2 and [Tod17b, Theorem 1.3], as explained in [Tod17b]. We can also obtain that M X,H (v) has symplectic singularities for arbitrary polarizations H, by combining Theorem 1.2 and [BS16, Proposition 1.2], which generalizes slightly [KLS06, Theorem 6.2].…”

Section: Introductionmentioning

confidence: 91%

Formality conjecture for K3 surfaces

Budur¹,

Zhang²

2019

Compositio Math.

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We give a proof of the formality conjecture of Kaledin and Lehn: on a complex projective K3 surface, the DG algebra RHom q (F, F ) is formal for any sheaf F polystable with respect to an ample line bundle. Our main tool is the uniqueness of the DG enhancement of the bounded derived category of coherent sheaves. We also extend the formality result to derived objects that are polystable with respect to a generic Bridgeland stability condition.

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Section: Introductionmentioning

confidence: 91%

Formality conjecture for K3 surfaces

Budur¹,

Zhang²

2019

Compositio Math.

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“…The conceptual payoff for reproving the theorem is that the second time we prove the theorem, we are able to make the isomorphism underlying the integrality theorem canonical -this is a key step towards proving the integrality theorem for the category of coherent sheaves on a 3-Calabi-Yau variety, for example. That this category locally looks like the category of representations for a quiver with potential is an exercise in deformation theory if one works formally locally, and is proved rigorously by Toda in the analytic topology [47]. The key step in proving the integrality conjecture for coherent sheaves on a 3-Calabi-Yau variety, then, is to prove it locally in a sufficiently canonical manner that the proof glues naturally.…”

Section: Theorem B (Cohomological Wall Crossing Theorem)mentioning

confidence: 99%

Cohomological Donaldson–Thomas theory of a quiver with potential and quantum enveloping algebras

2020

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This paper concerns the cohomological aspects of Donaldson-Thomas theory for Jacobi algebras and the associated cohomological Hall algebra, introduced by Kontsevich and Soibelman. We prove the Hodge-theoretic categorification of the integrality conjecture and the wall crossing formula, and furthermore realise the isomorphism in both of these theorems as Poincaré-Birkhoff-Witt isomorphisms for the associated cohomological Hall algebra.We do this by defining a perverse filtration on the cohomological Hall algebra, a result of the "hidden properness" of the semisimplification map from the moduli stack of semistable representations of the Jacobi algebra to the coarse moduli space of polystable representations.This enables us to construct a degeneration of the cohomological Hall algebra, for generic stability condition and fixed slope, to a free supercommutative algebra generated by a mixed Hodge structure categorifying the BPS invariants. As a corollary of this construction we furthermore obtain a Lie algebra structure on this mixed Hodge structure -the Lie algebra of BPS invariants -for which the entire cohomological Hall algebra can be seen as the positive part of a Yangian-type quantum group.

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“…The stack M σ (v) is a finite type open substack of M X (β). Indeed, the stack M σ (v) is constructed as a GIT quotient stack (see [Tod,Lemma 7.4]), hence we have the projective coarse moduli space M σ (v) together with the natural morphism…”

Section: Generalized Gv Invariantsmentioning

confidence: 99%

“…The above argument shows that the identity (5.7) holds at U ∩ {d(tr W ) = 0} for a sufficiently small analytic open subset 0 ∈ U ⊂ Ext 1 (E, E). By replacing V ⊂ M Q ( m) if necessary, we can assume that π −1 Q (V ) ⊂ G • U (see [Tod,Lemma 5.1]). Then by the G-invariance of both sides of (5.7), we have the identity (5.7) on {d(tr W ) = 0} ⊂ π −1 Q (V ).…”

Section: Appendix a Comparison Of D-critical Structuresmentioning

confidence: 99%

Gopakumar-Vafa invariants and wall-crossing

Toda¹

2017

Preprint

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In this paper, we generalize a mathematical definition of Gopakumar-Vafa (GV) invariants on Calabi-Yau 3-folds introduced by Maulik and the author, using an analogue of BPS sheaves introduced by Davison-Meinhardt on the coarse moduli spaces of one dimensional twisted semistable sheaves with arbitrary holomorphic Euler characteristics. We show that our generalized GV invariants are independent of twisted stability conditions, and conjecture that they are also independent of holomorphic Euler characteristics, so that they define the same GV invariants. As an application, we will show the flop transformation formula of GV invariants.

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Moduli stacks of semistable sheaves and representations of Ext-quivers

Cited by 3 publications

References 19 publications

Formality conjecture for K3 surfaces

Formality conjecture for K3 surfaces

Cohomological Donaldson–Thomas theory of a quiver with potential and quantum enveloping algebras

Gopakumar-Vafa invariants and wall-crossing

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