Nested Hilbert schemes on surfaces: Virtual fundamental class

Gholampour, Amin; Sheshmani, Artan; Yau, Shing–Tung

doi:10.48550/arxiv.1701.08899

Cited by 12 publications

(44 citation statements)

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“…Let N := N H S (2, c 1 , c 2 ) and let M mon ⊂ N C * be the monopole branch discussed in the introduction. Gholampour-Thomas [GT1] (see also [GSY1]) prove that the components of M mono are isomorphic to…”

Section: Monopole Contribution and Nested Hilbert Schemesmentioning

confidence: 98%

Verlinde formulae on complex surfaces: K-theoretic invariants

Göttsche¹,

Kool²,

Williams³

2019

Preprint

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We conjecture a Verlinde type formula for the moduli space of Higgs sheaves on a surface with a holomorphic 2-form. The conjecture specializes to a Verlinde formula for the moduli space of sheaves. Our formula interpolates between K-theoretic Donaldson invariants studied by the first named author and Nakajima-Yoshioka and K-theoretic Vafa-Witten invariants introduced by Thomas and also studied by the first and second named authors. We verify our conjectures in many examples (e.g. on K3 surfaces).

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Section: Monopole Contribution and Nested Hilbert Schemesmentioning

confidence: 98%

Verlinde formulae on complex surfaces: K-theoretic invariants

Göttsche¹,

Kool²,

Williams³

2019

Preprint

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“…, λ k ) of r. We will study the contribution of the locus P ⊥ 1 r = P ⊥ (1,...,1) of vertical Joyce-Song pairs to the Vafa-Witten invariants. Using [GSY18] and [GT19], we will construct the schemes P ⊥ 1 r and their virtual classes directly in Sections 2-4. 1.4.…”

Section: We Will Writementioning

confidence: 99%

“…β of the perfect obstruction theory on the nested Hilbert scheme S n β . It follows directly by the description of [GSY18] that it is given by given by the dual of a cone on (5.1)…”

Section: Stabilitymentioning

confidence: 99%

Vertical Vafa-Witten invariants

Laarakker¹

2019

Preprint

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We show that vertical contributions to (possibly semistable) Tanaka-Thomas-Vafa-Witten invariants are well defined for surfaces with pg(S) > 0, partially proving conjectures of [TT17b] and [Tho18a]. Moreover, we show that such contributions are computed by the same tautological integrals as in the stable case, which we studied in [Laa18]. Using the work of Kiem and Li, we show that stability of universal families of vertical Joyce-Song pairs is controlled by cosections of the obstruction sheaves of such families. Contents1. Introduction 1 1.1. Joyce-Song pairs 1 1.2. Vafa-Witten invariants 2 1.3. The fixed locus 3 1.4. Results 3 1.5. Acknowledgement 4 2. A tautological family of Joyce-Song pairs 4 3. The virtual class 6 4. Localising the virtual class 9 5. Stability 10 6. Comparison to the VW moduli space 13 7. The unrefined case 15 8. The integrals 17 9. Refined invariants 20 References 21

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“…The nested Hilbert schemes are studied in [5], in which a perfect obstruction theory is constructed. Write I [ni] for the universal ideal sheaf on S [ni] × S, and let D i → Hilb βi (S) × S be the universal curve with class β i .…”

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confidence: 99%

Monopole contributions to refined Vafa-Witten invariants

Laarakker

2018

Preprint

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We study the monopole contribution to the refined Vafa-Witten invariant, recently defined in [17]. We apply the results of [7] to prove a universality result for the generating series of contributions of Higgs pairs with 1-dimensional weight spaces. For prime rank, these account for the entire monopole contribution by a theorem of Thomas. We use toric computations to determine part of the generating series, and find agreement with the conjectures of [11] for rank 2 and 3.

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Nested Hilbert schemes on surfaces: Virtual fundamental class

Cited by 12 publications

References 0 publications

Verlinde formulae on complex surfaces: K-theoretic invariants

Verlinde formulae on complex surfaces: K-theoretic invariants

Vertical Vafa-Witten invariants

Monopole contributions to refined Vafa-Witten invariants

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