Curve counting invariants for crepant resolutions

Bryan, Jenny; Steinberg, David J.

doi:10.1090/tran/6377

Cited by 16 publications

(37 citation statements)

References 48 publications

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“…If s ∈ D then K is supported over F n = Spec O X /π * m n s for some n ≫ 0 where m s is the ideal sheaf of s ∈ S. Let f ∈ O X be the local equation of D near s, and let f be its image in O S /m n s . Consider the exact sequence (11) 0 → (f ) → O S /m n s → O S /(m n s , f ) → 0 . By flatness of π, (11) remains exact under pullback: (12) 0 → π * (f ) → O X /π * m n s → O X /π * (m n s , f ) → 0 .…”

Section: Proof Of Proposition 4 Assume a Complex I • Satisfies (A-d)mentioning

confidence: 99%

“…The definition of π-stable pairs is similar in philosophy to the modification of stable pairs by Bryan-Steinberg [11] for a crepant resolution X → Y that contracts an exceptional curve. While for BS-pairs we allow a pair O X → F to have 1-dimensional cokernel along the exceptional curve, here we allow a π-stable pair to have 1-dimensional cokernel supported on arbitrary fibers of the fibration.…”

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confidence: 99%

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Curve counting on elliptic Calabi–Yau threefolds via derived categories

Oberdieck

Shen

2019

J. Eur. Math. Soc.

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We prove the elliptic transformation law of Jacobi forms for the generating series of Pandharipande-Thomas invariants of an elliptic Calabi-Yau 3-fold over a reduced class in the base. This proves part of a conjecture by Huang, Katz, and Klemm. For the proof we construct an involution of the derived category and use wall-crossing methods. We express the generating series of PT invariants in terms of low genus Gromov-Witten invariants and universal Jacobi forms.As applications we prove new formulas and recover several known formulas for the PT invariants of K3 × E, abelian 3-folds, and the STUmodel. We prove that the generating series of curve counting invariants for K3 × E with respect to a primitive class on the K3 is a quasi-Jacobi form of weight -10. This provides strong evidence for the Igusa cusp form conjecture.Let Z H (q, t) = PT H (q, t)/PT 0 (q, t). Then Theorem 1 can be rewritten asBy [5,37] every series n∈Z P n,H+dF q n is the Laurent expansion of a rational function in q invariant under the variable change q → q −1 . Considering Z H (q, t) as an element in Q(q)[[t]], we therefore also have Z H (q −1 , t) = Z H (q, t) .

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Section: Proof Of Proposition 4 Assume a Complex I • Satisfies (A-d)mentioning

confidence: 99%

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confidence: 99%

Curve counting on elliptic Calabi–Yau threefolds via derived categories

Oberdieck

Shen

2019

J. Eur. Math. Soc.

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“…Bryan-Steinberg (BS) theory [5] provides a geometric approach to this normalisation. Roughly, whereas PT theory allows the cokernel Q to be 0-dimensional, BS theory allows Q to develop 1-dimensional components supported only on .…”

Section: 25mentioning

confidence: 99%

Quasimaps and stable pairs

Liu

2021

Forum of Mathematics, Sigma

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We prove an equivalence between the Bryan-Steinberg theory of $\pi $ -stable pairs on $Y = \mathcal {A}_{m-1} \times \mathbb {C}$ and the theory of quasimaps to $X = \text{Hilb}(\mathcal {A}_{m-1})$ , in the form of an equality of K-theoretic equivariant vertices. In particular, the combinatorics of both vertices are described explicitly via box counting. Then we apply the equivalence to study the implications for sheaf-counting theories on Y arising from 3D mirror symmetry for quasimaps to X, including the Donaldson-Thomas crepant resolution conjecture.

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“…As in [1] and [7], the coherent sheaves O C (−1), O C (−2) are not perverse, hence the exact sequences (12) and (13) do not define exact sequences in Per(X /Y). But the shifted ones O C (−1) [1], O C (−2) [1] are perverse sheaves, and we have:…”

Section: The Derived Equivalencementioning

confidence: 99%

“…In [19], Calabrese proves part of the conjecture for Calabi-Yau threefold stacks satisfying the HL conditions, using similar method of Hall algebra identities in [18]. Note that Bryan and Steinberg [13] also prove partial result of the crepant resolution conjecture for the DTinvariants. We hope that our study of orbifold flop may shed more light on the crepant resolution conjecture.…”

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confidence: 93%

Donaldson–Thomas invariants of Calabi–Yau orbifolds under flops

Jiang¹

2018

Illinois J. Math.

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We study the Donaldson-Thomas type invariants for the Calabi-Yau threefold Deligne-Mumford stacks under flops. A crepant birational morphism between two smooth Calabi-Yau threefold Deligne-Mumford stacks is called an orbifold flop if the flopping locus is the quotient of weighted projective lines by a cyclic group action. We prove that the Donaldson-Thomas invariants are preserved under orbifold flops. CONTENTSYUNFENG JIANG Proof of Theorem 1.3 32 6. Discussion on the Hard Lefschetz condition. 32 33 References 33 DONALDSON-THOMAS INVARIANTS UNDER FLOPS 3 Definition 1.1. ([43]) A stable pair(1) dim(F) ≤ 1 and F is pure;(2) s has zero-dimensional cokernel.The moduli scheme PT n (X, β) of stable pairs with fixing topological data [F] = β ∈ H 2 (X, Z), χ(F) = n is a scheme and the PT-invariant is defined by PT n,β (X) = χ(PT n (X, β), ν PT ), where ν PT is the Behrend function on PT n (X, β). Both DT-invariants and PT-invariants are curve counting invariants of X; The famous DT/PTcorrespondence conjecture in [43] equates these two invariants in terms of partition functions.The conjecture was proved by Bridgeland [9], and Toda [47] using the wall crossing idea, under which the DT-moduli space and the PT moduli space correspond to different (limit) stability conditions in Bridgeland's space of stability conditions.We pay more attention to Bridgeland's method for the proof. In [9] Bridgeland uses some Hall algebra identities in the motivic Hall algebra H(A) of the abelian category of coherent sheaves A = Coh(X); such that the DT-moduli space and the PT-moduli space are both elements in the Hall algebra. Then applying the integration map as in [33], [10] Bridgeland gets the DT/PT-correspondence.The same idea works for threefold flops. Let

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Curve counting invariants for crepant resolutions

Cited by 16 publications

References 48 publications

Curve counting on elliptic Calabi–Yau threefolds via derived categories

Curve counting on elliptic Calabi–Yau threefolds via derived categories

Quasimaps and stable pairs

Donaldson–Thomas invariants of Calabi–Yau orbifolds under flops

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