A proof of the Donaldson-Thomas crepant resolution conjecture

Beentjes, Sjoerd Viktor; Calabrese, John; Rennemo, Jørgen Vold

doi:10.48550/arxiv.1810.06581

Cited by 4 publications

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“…In [57], the author also gives the definition of the absolute and relative orbifold DT theory (see also [8,17] for the prior definition with Calabi-Yau condition) and derives the corresponding degeneration formulas. In [33], the author constructs moduli spaces of orbifold PT stable pairs and defines virtual fundamental classes which are integrated to give the absolute orbifold PT invariants for 3-dimensional smooth projective Deligne-Mumford stacks (see also [5] for the prior definition in the case of Calabi-Yau 3-orbifolds). However, it is naturally expected to have the relative orbifold PT theory and the corresponding degeneration formula.…”

Section: Introductionmentioning

confidence: 99%

Relative orbifold Pandharipande-Thomas theory and the degeneration formula

Lin

2021

Preprint

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We construct relative moduli spaces of semistable pairs on a family of projective Deligne-Mumford stacks. We define moduli stacks of stable orbifold Pandharipande-Thomas pairs on stacks of expanded degenerations and pairs, and then show they are separated and proper Deligne-Mumford stacks of finite type. As an application, we present the degeneration formula for the absolute and relative orbifold Pandharipande-Thomas invariants. Contents 1. Introduction 1 2. Preliminary 5 2.1. Projective Deligne-Mumford stacks 5 2.2. Stacks of expanded degenerations and pairs 7 2.3. Admissibility of sheaves and its numerical criterion 12 3. Relative moduli spaces of semistable pairs 13 3.1. Semistable pairs on the family of projective Deligne-Mumford stacks 13 3.2. Boundedness of the family of semistable pairs 14 3.3. Construction of relative moduli spaces of semistable pairs 16 4. Perfect relative obstruction theory and virtual fundamental class 20 5. Moduli of stable orbifold Pandharipande-Thomas pairs 21 5.1. Stable orbifold PT pairs 21 5.2. Moduli of stable orbifold PT pairs 23 5.3. Moduli of stable orbifold PT pairs with fixed Hilbert homomorphism 24 5.4. Properties of the moduli of stable orbifold PT pairs 25 6. Degeneration formula 29 6.1. Decomposition of central fibers 29 6.2. Absolute and relative orbifold Pandharipande-Thomas theory 32 6.3. Cycle version of degeneration formula 35 6.4. Numerical version of degeneration formula 37 References 39

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Section: Introductionmentioning

confidence: 99%

Relative orbifold Pandharipande-Thomas theory and the degeneration formula

Lin

2021

Preprint

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“…It is natural and expected to have an orbifold PT theory together with the orbifold GW/PT or DT/PT correspondence. In [7], the authors follow Toda's method [63] of applying the notion of a torsion pair to obtain a stacky version of PT stable pairs, and then combine the motivic Hall algebra (cf. [11]) and Behrend's constructible function [8] to define the orbifold PT invariants for smooth projective Calabi-Yau 3-orbifolds.…”

Section: Introductionmentioning

confidence: 99%

“…Definition 5.23) for smooth projective Deligne-Mumford stacks without Calabi-Yau condition. When X is a 3-dimensional Calabi-Yau orbifold, our definition of orbifold PT invariants of X is corresponding to the one in [7] (cf. Definition 5.25 and Remark 5.26) due to the application of [8,Theorem 4.18] for the moduli space with a symmetric perfect obstruction theory.…”

Section: Introductionmentioning

confidence: 99%

“…Definition 5.23) as in [63,10,47,48,49]. And it is natural to investigate the conjecture of orbifold GW/PT or DT/PT correspondence as in [68,69,7], or more generally the stacky version of GW/PT or DT/PT correspondence proved in [41,50,44,63,10]. One may also consider the stacky version of higher rank PT stable pairs in [60].…”

Section: Introductionmentioning

confidence: 99%

“…We define Pandharipande-Thomas invariants of X corresponding to P and β as followsPT(O X , P, δ) := χ(M s X /C (O X , P, δ), ν M s ) = deg([M s X /C (O X , P, δ)] vir ), PT(O X , β, δ) := χ(M s X /C (O X , β, δ), ν M s β ) = deg([M s X /C (O X , β, δ)] vir ).Remark 5.26. When X is a 3-dimensional Calabi-Yau orbifold, the invariant PT(O X , β, δ) in the above definition is corresponding to PT(X ) (β,0) defined in[7, Section 1 (1.4)].…”

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confidence: 99%

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Moduli spaces of semistable pairs on projective Deligne-Mumford stacks

Lin¹

2020

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We generalize the construction of a moduli space of semistable pairs parametrizing isomorphism classes of morphisms from a fixed coherent sheaf to any sheaf with fixed Hilbert polynomial under a notion of stability to the case of projective Deligne-Mumford stacks. We study the deformation and obstruction theories of stable pairs, and then prove the existence of virtual fundamental classes for some cases of dimension two and three. This leads to a definition of Pandharipande-Thomas invariants on three-dimensional smooth projective Deligne-Mumford stacks.

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Rank $r$ DT theory from rank $0$

Feyzbakhsh¹,

Thomas²

2021

Preprint

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Fix a Calabi-Yau 3-fold X satisfying the Bogomolov-Gieseker conjecture of Bayer-Macrì-Toda, such as the quintic 3-fold. We express Joyce's generalised DT invariants counting Gieseker semistable sheaves of any rank r ≥ 1 on X in terms of those counting sheaves of rank 0 and pure dimension 2.The basic technique is to reduce the ranks of sheaves by replacing them by the cokernels of their Mochizuki/Joyce-Song pairs and then use wall crossing to handle their stability.It has long been speculated that the higher rank or "nonabelian" DT theory of a Calabi-Yau 3-fold might be governed by its rank one "abelian" theory. 1 This would be a 6dimensional analogue of the correspondence between nonabelian Donaldson theory and abelian Seiberg-Witten theory for smooth 4-manifolds. Combined with the MNOP conjecture [MNOP], now proved for most Calabi-Yau 3-folds [PP], it would express DT theory in any rank entirely in terms of Gromov-Witten invariants. Contents 1. Weak stability conditions 4 2. Semistable objects are sheaves 6 3. Destabilising objects are sheaves 13 4. Enumerative invariants and wall crossing formulae 21 5. Crossing the walls 28 Appendix A. Joyce-Song pairs 31 Appendix B. Rank 2 simplifications 33 Appendix C. Moduli stacks 36 References 39Plan. We work on a smooth complex projective threefold X. In Section 1 we review the weak stability conditions and Bogomolov-Gieseker inequality of [BMT, BMS]. Section 2 proves that for such stability conditions, semistable objects E ∈ D(X) of class v n are always sheaves. Moreover in Section 3 we show that their destabilising objects are also sheaves, with one exception -on the Joyce-Song wall, E can be destabilised by an exact triangleexpressing it in terms of a Joyce-Song pair (2) up to tensoring by some T ∈ Pic 0 (X).From Section 4 we assume X is Calabi-Yau. We review the invariants counting its (weak) semistable sheaves and their wall crossing formulae in Section 4. Section 5 gives the proof of Theorem 1. It hides the role of Joyce-Song stable pairs so we explain their relevance in Appendix A, and stronger results in rank 2 in Appendix B. Neither is necessary for the rest of the paper. Appendix C uses work of Piyaratne-Toda to check that moduli stacks of weak semistable objects are algebraic of finite type.Outlook. Up until Section 4 all of our results apply to any 3-fold satisfying the Bogomolov-Gieseker inequality. From Section 4 we restrict attention to Calabi-Yau 3-folds only to apply the wall crossing formula. Joyce is currently developing a a wall crossing formula that should apply to Fano 3-folds [GJT, Conjecture 4.2]; we would then expect our results to prove a version of Theorem 1 with insertions.On Calabi-Yau 3-folds with h 1 (O X ) > 0 the locally free action of Jac(X) on moduli spaces of rank r > 0 sheaves forces J(v) = 0. So from Section 4 we assume h 1 (O X ) = 0 without loss of generality. But it would be interesting to extend our results to counts of sheaves of fixed determinant when h 1 (O X ) > 0; this should not be too hard because in this case the wall cro...

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A proof of the Donaldson-Thomas crepant resolution conjecture

Cited by 4 publications

References 0 publications

Relative orbifold Pandharipande-Thomas theory and the degeneration formula

Relative orbifold Pandharipande-Thomas theory and the degeneration formula

Moduli spaces of semistable pairs on projective Deligne-Mumford stacks

Rank $r$ DT theory from rank $0$

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