An Application of Wall-Crossing to Noether–Lefschetz Loci

Feyzbakhsh, Soheyla; Thomas, Robert Paul; Voisin, Claire

doi:10.1093/qmathj/haaa022

Cited by 5 publications

(3 citation statements)

References 18 publications

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“…This paper, its predecessor [FT21a] and its sequel [Fey22] use methods pioneered by Yukinobu Toda. In [Tod12] he also studied 2-dimensional sheaves on threefolds X satisfying the Bogomolov-Gieseker inequality, under the additional assumption that X is Calabi-Yau with Pic X = Z.…”

Section: Relationship To the Work Of Todamentioning

confidence: 99%

“…This would give useful relations between enumerative invariants counting sheaves, but it very rarely works due to stability issues. This is one of a series of papers [Fey22,FT21a,FT21b,FT21c] showing it can be made to work -modulo the wall-crossing formulae required to move to a stability condition for which such cokernels are stableon a threefold X satisfying (a weakening of) Bayer-Macrì-Toda's Bogomolov-Gieseker conjecture [BMT14]. When X is Calabi-Yau, this gives relations between invariants involving unwieldy formulae.…”

Section: Introductionmentioning

confidence: 99%

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Curve counting and S-duality

Feyzbakhsh¹,

Thomas²

2023

Épijournal De Géométrie Algébrique

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We work on a projective threefold $X$ which satisfies the Bogomolov-Gieseker conjecture of Bayer-Macr\`i-Toda, such as $\mathbb P^3$ or the quintic threefold. We prove certain moduli spaces of 2-dimensional torsion sheaves on $X$ are smooth bundles over Hilbert schemes of ideal sheaves of curves and points in $X$. When $X$ is Calabi-Yau this gives a simple wall crossing formula expressing curve counts (and so ultimately Gromov-Witten invariants) in terms of counts of D4-D2-D0 branes. These latter invariants are predicted to have modular properties which we discuss from the point of view of S-duality and Noether-Lefschetz theory.

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Section: Relationship To the Work Of Todamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Curve counting and S-duality

Feyzbakhsh¹,

Thomas²

2023

Épijournal De Géométrie Algébrique

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“…See also [Bay18,BL17,Fey19,Fey20,FL18,FT19,FT20] for other interesting applications of the wall-crossing in tilt-stability. 1.5.…”

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confidence: 99%

On the Bogomolov-Gieseker inequality for hypersurfaces in the projective spaces

Koseki¹

2020

Preprint

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An effective restriction theorem via wall-crossing and Mercat’s conjecture

Feyzbakhsh

2022

Math. Z.

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We prove an effective restriction theorem for stable vector bundles E on a smooth projective variety: $$E|_D$$ E | D is (semi)stable for all irreducible divisors $$D \in |kH|$$ D ∈ | k H | for all k greater than an explicit constant. As an application, we show that Mercat’s conjecture in any rank greater than 2 fails for curves lying on K3 surfaces. Our technique is to use wall-crossing with respect to (weak) Bridgeland stability conditions which we also use to reprove Camere’s result on slope stability of the tangent bundle of $${\mathbb {P}}^n$$ P n restricted to a K3 surface.

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An Application of Wall-Crossing to Noether–Lefschetz Loci

Cited by 5 publications

References 18 publications

Curve counting and S-duality

Curve counting and S-duality

On the Bogomolov-Gieseker inequality for hypersurfaces in the projective spaces

An effective restriction theorem via wall-crossing and Mercat’s conjecture

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