$G$-invariant Hilbert Schemes on Abelian Surfaces and Enumerative Geometry of the Orbifold Kummer Surface

Pietromonaco, Stephen

doi:10.48550/arxiv.2011.14020

Cited by 2 publications

(3 citation statements)

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“…is a holomorphic modular form provided that X is a K3 surface (a similar statement for abelian surfaces was obtained recently in [10]). Note however that in Corollary 1.4 we do not take reciprocal.…”

Section: Introductionsupporting

confidence: 59%

Rigid ideal sheaves and modular forms

Gyenge¹

2021

Preprint

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Let X be a complex smooth quasi-projective surface acted upon by a finite group G such that the quotient X/G has singularities only of ADE type. We obtain an explicit expression for the generating series of the Euler characteristics of the zero-dimensional components in the moduli space of zero-dimensional subschemes on X invariant under the action of G. We show that this generating series (up to a suitable rational power of the formal variable) is a holomorphic modular form.

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Section: Introductionsupporting

confidence: 59%

Rigid ideal sheaves and modular forms

Gyenge¹

2021

Preprint

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“…The case where G is a cyclic group was proved in [BO18]. An analogous result for the case where X is an Abelian surface acted on symplectically by a finite group G has been recently proven by Pietromonaco [Pie20].…”

Section: The Main Resultsmentioning

confidence: 71%

“…This generalizes the famous Yau-Zaslow formula [YZ96] in the case where G is the trivial group. The precise nature between the virtual count and the actual count is expected to be subtle for the case of general G. This has been recently explored in [Zha19] and also in the case of G acting on an Abelian surface in [Pie20].…”

Section: Consequences Of the Main Resultsmentioning

confidence: 99%

$G$-fixed Hilbert schemes on $K3$ surfaces, modular forms, and eta products

Bryan¹,

Gyenge²

2022

Épijournal De Géométrie Algébrique

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Let $X$ be a complex $K3$ surface with an effective action of a group $G$ which preserves the holomorphic symplectic form. Let $$ Z_{X,G}(q) = \sum_{n=0}^{\infty} e\left(\operatorname{Hilb}^{n}(X)^{G} \right)\, q^{n-1} $$ be the generating function for the Euler characteristics of the Hilbert schemes of $G$-invariant length $n$ subschemes. We show that its reciprocal, $Z_{X,G}(q)^{-1}$ is the Fourier expansion of a modular cusp form of weight $\frac{1}{2} e(X/G)$ for the congruence subgroup $\Gamma_{0}(|G|)$. We give an explicit formula for $Z_{X,G}$ in terms of the Dedekind eta function for all 82 possible $(X,G)$. The key intermediate result we prove is of independent interest: it establishes an eta product identity for a certain shifted theta function of the root lattice of a simply laced root system. We extend our results to various refinements of the Euler characteristic, namely the Elliptic genus, the Chi-$y$ genus, and the motivic class.

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$G$-invariant Hilbert Schemes on Abelian Surfaces and Enumerative Geometry of the Orbifold Kummer Surface

Cited by 2 publications

References 0 publications

Rigid ideal sheaves and modular forms

Rigid ideal sheaves and modular forms

$G$-fixed Hilbert schemes on $K3$ surfaces, modular forms, and eta products

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