Virtual classes and virtual motives of Quot schemes on threefolds

Ricolfi, Andrea T.

doi:10.1016/j.aim.2020.107182

Cited by 19 publications

(41 citation statements)

References 25 publications

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“…1 ∈ Z can be computed via the Graber-Pandharipande virtual localisation formula [27]. The next result confirms (in the toric case) a prediction [59,Conj. 3.5] for their generating function.…”

Section: Global Geometrysupporting

confidence: 75%

“…The subtle structure of the DT moduli spaces is most evident in the local case; that is, when the theory is applied to the simplest Calabi-Yau 3-fold of all, namely, the affine space A 3 . See [6,20] for the rank 1 motivic DT theory of A 3 and [59] for a higher rank version. This article solves the K-theoretic Donaldson-Thomas theory of points of A 3 .…”

Section: Overviewmentioning

confidence: 99%

“…So far we have only discussed results concerning local geometry. When is a projective toric 3-fold and is an equivariant exceptional locally free sheaf, by [59,Thm. A] there is a 0-dimensional torus equivariant (cf. Proposition 9.2) perfect obstruction theory on Quot ( , ).…”

Section: Global Geometrymentioning

confidence: 99%

“…Virtual classes of dimension 0 of Quot schemes (for locally free sheaves) on projective 3-folds are constructed under certain assumptions in [59]. In this article we mainly work with the virtual class on Quot A 3 ( ⊕ , ) coming from the critical obstruction theory found in [4].…”

Section: Related Work On Virtual Invariants Of Quot Schemesmentioning

confidence: 99%

“…Degree 0 DT invariants of various flavours have been computed in [38,39,8,28,45,59,6,15]. This article is concerned with the general theory of Quot schemes and hence in the (virtual) enumeration of 0-dimensional quotients of locally free sheaves on 3-folds.…”

Section: Classical Enumerative Invariantsmentioning

confidence: 99%

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Higher rank K-theoretic Donaldson-Thomas Theory of points

Fasola

Monavari

Ricolfi

2021

Forum of Mathematics, Sigma

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We exploit the critical structure on the Quot scheme $\text {Quot}_{{{\mathbb {A}}}^3}({\mathscr {O}}^{\oplus r}\!,n)$ , in particular the associated symmetric obstruction theory, in order to study rank r K-theoretic Donaldson-Thomas (DT) invariants of the local Calabi-Yau $3$ -fold ${{\mathbb {A}}}^3$ . We compute the associated partition function as a plethystic exponential, proving a conjecture proposed in string theory by Awata-Kanno and Benini-Bonelli-Poggi-Tanzini. A crucial step in the proof is the fact, nontrival if $r>1$ , that the invariants do not depend on the equivariant parameters of the framing torus $({{\mathbb {C}}}^\ast )^r$ . Reducing from K-theoretic to cohomological invariants, we compute the corresponding DT invariants, proving a conjecture of Szabo. Reducing further to enumerative DT invariants, we solve the higher rank DT theory of a pair $(X,F)$ , where F is an equivariant exceptional locally free sheaf on a projective toric $3$ -fold X. As a further refinement of the K-theoretic DT invariants, we formulate a mathematical definition of the chiral elliptic genus studied in physics. This allows us to define elliptic DT invariants of ${{\mathbb {A}}}^3$ in arbitrary rank, which we use to tackle a conjecture of Benini-Bonelli-Poggi-Tanzini.

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Section: Global Geometrysupporting

confidence: 75%

Section: Overviewmentioning

confidence: 99%

Section: Global Geometrymentioning

confidence: 99%

Section: Related Work On Virtual Invariants Of Quot Schemesmentioning

confidence: 99%

Section: Classical Enumerative Invariantsmentioning

confidence: 99%

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Higher rank K-theoretic Donaldson-Thomas Theory of points

Fasola

Monavari

Ricolfi

2021

Forum of Mathematics, Sigma

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Framed motivic Donaldson–Thomas invariants of small crepant resolutions

Cazzaniga

Ricolfi

2022

Mathematische Nachrichten

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For an arbitrary integer r≥1$r\ge 1$, we compute r‐framed motivic DT and PT invariants of small crepant resolutions of toric Calabi–Yau 3‐folds, establishing a “higher rank” version of the motivic DT/PT wall‐crossing formula. This generalises the work of Morrison and Nagao. Our formulae, in particular their relationship with the r=1$r=1$ theory, fit nicely in the current development of higher rank refined DT invariants.

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Degree 0 DT Invariants of a Local Calabi–Yau 3-Fold

Ricolfi

2022

SISSA Springer Series

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Virtual classes and virtual motives of Quot schemes on threefolds

Cited by 19 publications

References 25 publications

Higher rank K-theoretic Donaldson-Thomas Theory of points

Higher rank K-theoretic Donaldson-Thomas Theory of points

Framed motivic Donaldson–Thomas invariants of small crepant resolutions

Degree 0 DT Invariants of a Local Calabi–Yau 3-Fold

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