Holomorphic anomaly equations for the Hilbert scheme of points of a K3 surface

Oberdieck, Georg

doi:10.48550/arxiv.2202.03361

Cited by 3 publications

(4 citation statements)

References 77 publications

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“…We mention here the following basic qualitative property of the invariants of S [n] , which gives an affirmative answer to [47,Question 1.4] for (E × C) [n] .…”

mentioning

confidence: 96%

“…For K3 surfaces S the reduced 1 quantum cohomology of S [2] was computed in [46], and a conjectural formula for quantum multiplication with divisor classes for any S [n] was given in [46,13]. In [47] the structure constants of the reduced quantum product of K3 [n] were shown to be quasi-Jacobi forms. This determines the reduced quantum cohomology of K3 [n] up to finitely many coefficients.…”

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confidence: 99%

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A Lie algebra action on the Chow ring of the Hilbert scheme of points of a K3 surface

Oberdieck

2021

Comment. Math. Helv.

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We determine the quantum multiplication with divisor classes on the Hilbert scheme of points on an elliptic surface S → Σ for all curve classes which are contracted by the induced fibration S [n] → Σ [n] . The formula is expressed in terms of explicit operators on Fock space. The structure constants are meromorphic quasi-Jacobi forms of index 0. Combining with work of Hu-Li-Qin, this determines the quantum multiplication with divisors on the Hilbert scheme of elliptic surfaces with pg(S) > 0. We also determine the equivariant quantum multiplication with divisor classes for the Hilbert scheme of points on the product E × C.The proof of our formula is based on Nesterov's Hilb/PT wall-crossing, a newly established GW/PT correspondence for the product of an elliptic surface times a curve, and new computations in the Gromov-Witten theory of an elliptic curve.

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“…We mention here the following basic qualitative property of the invariants of S [n] , which gives an affirmative answer to [47,Question 1.4] for (E × C) [n] .…”

mentioning

confidence: 96%

mentioning

confidence: 99%

A Lie algebra action on the Chow ring of the Hilbert scheme of points of a K3 surface

Oberdieck

2021

Comment. Math. Helv.

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“…This completes the proof of the Igusa cusp form conjecture and provides an expression for GW invariants of 𝑆 [𝑛] associated to E. We refer to Section 4.2 for more details about this conjecture. In a similar vein, in [Obe22], a holomorphic anomaly equation is established for 𝑆 [𝑛] for genus-0 GW invariants with at most three markings. The proof crucially uses the quisimap wall-crossing, which relates genus-0 GW invariants of 𝑆 [𝑛] to PT invariants of 𝑆 × P 1 and then to GW invariants of 𝑆 × P 1 by [Obe21a].…”

Section: Enumerative Geometry Of 𝑆mentioning

confidence: 99%

Quasimaps to moduli spaces of sheaves on a surface

Nesterov

2024

Forum of Mathematics, Sigma

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In this article, we study quasimaps to moduli spaces of sheaves on a $K3$ surface S. We construct a surjective cosection of the obstruction theory of moduli spaces of $\epsilon $ -stable quasimaps. We then establish reduced wall-crossing formulas which relate the reduced Gromov–Witten theory of moduli spaces of sheaves on S and the reduced Donaldson–Thomas theory of $S\times C$ , where C is a nodal curve. As applications, we prove the Hilbert-schemes part of the Igusa cusp form conjecture; higher-rank/rank-one Donaldson–Thomas correspondence with relative insertions on $S\times C$ , if $g(C)\leq 1$ ; Donaldson–Thomas/Pandharipande–Thomas correspondence with relative insertions on $S\times \mathbb {P}^1$ .

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“…We will reinterpret the lemma in terms of Jacobi forms in [44]. See also [1] for a parallel discussion in the case of K3 surfaces.…”

Section: Hilbert Schemes Of Elliptic K3 Surfacesmentioning

confidence: 99%

Gromov–Witten theory and Noether–Lefschetz theory for holomorphic-symplectic varieties

Oberdieck¹

2022

Forum of Mathematics, Sigma

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We use Noether–Lefschetz theory to study the reduced Gromov–Witten invariants of a holomorphic-symplectic variety of $K3^{[n]}$ -type. This yields strong evidence for a new conjectural formula that expresses Gromov–Witten invariants of this geometry for arbitrary classes in terms of primitive classes. The formula generalizes an earlier conjecture by Pandharipande and the author for K3 surfaces. Using Gromov–Witten techniques, we also determine the generating series of Noether–Lefschetz numbers of a general pencil of Debarre–Voisin varieties. This reproves and extends a result of Debarre, Han, O’Grady and Voisin on Hassett–Looijenga–Shah (HLS) divisors on the moduli space of Debarre–Voisin fourfolds.

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Holomorphic anomaly equations for the Hilbert scheme of points of a K3 surface

Cited by 3 publications

References 77 publications

A Lie algebra action on the Chow ring of the Hilbert scheme of points of a K3 surface

A Lie algebra action on the Chow ring of the Hilbert scheme of points of a K3 surface

Quasimaps to moduli spaces of sheaves on a surface

Gromov–Witten theory and Noether–Lefschetz theory for holomorphic-symplectic varieties

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