## Abstract

For an Abelian surface *A* with a symplectic action by a finite group *G*, one can define the partition function for *G*-invariant Hilbert schemes

We prove the reciprocal \(Z_{A,G}^{-1}\) is a modular form of weight \(\frac{1}{2}e(A/G)\) for the congruence subgroup \(\Gamma _{0}(|G|)\) and give explicit expressions in terms of eta products. Refined formulas for the \(\chi _{y}\)-genera of \(\text {Hilb}(A)^{G}\) are also given. For the group generated by the standard involution \(\tau : A \rightarrow A\), our formulas arise from the enumerative geometry of the orbifold Kummer surface \([A/\tau ]\). We prove that a virtual count of curves in the stack is governed by \(\chi _{y}(\text {Hilb}(A)^{\tau })\). Moreover, the coefficients of \(Z_{A, \tau }\) are true (weighted) counts of rational curves, consistent with hyperelliptic counts of Bryan, Oberdieck, Pandharipande, and Yin.

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## Notes

Note that \(G_{0}\) and \(G'\) are abstractly isomorphic. But we distinguish them because they are different groups acting on different spaces.

Throughout, we must handle \(d=0\) separately. In this case, choose the product Abelian surface \(A=E \times F\), with \(\beta _{0}\) the class of \(E \times \{\text {pt}\}\).

The definition of the BPS invariants by Maulik–Toda applies to Calabi–Yau threefold. But in the case of a local Calabi–Yau surface, the theory reduces to a theory of sheaves on the surface, see Sect. 4.1. Our results are therefore intrinsic to \([A/\tau ]\).

In [5, Sec. 5.4] what we are calling \({\mathsf {h}}_{d}(g)\) was denoted \(\textsf {h}_{g, \beta }^{A,{\text {Hilb}}}\).

This was equivalently carried out from a physics perspective in [21] by studying symmetry groups of certain nonlinear sigma models on the underlying real torus \(T^{4}\).

Here, \(M_{X}(0, 0, \beta , 1)\) is the moduli space of Simpson stable torsion sheaves on

*X*with generic polarization, and Chern character \((0,0,\beta ,1) \in H^{2*}(X, {\mathbb {Z}})\). Moreover, \({\text {Chow}}_{\beta }(X)\) is the Chow variety of effective curves in the class \(\beta \).See Remark 1.7.

Equivalently, \(\eta (q)\) is a modular form of weight \(\frac{1}{2}\) on the metaplectic double cover \(\text {Mp}_{2}({\mathbb {Z}})\) of \({\text {SL}}_{2}({\mathbb {Z}})\).

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## Acknowledgements

I would like to thank my advisor Jim Bryan for his guidance and many helpful suggestions throughout this project, as well as the anonymous referee for their important feedback.

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## Appendix A. Modular forms and eta products

### Appendix A. Modular forms and eta products

This appendix will be devoted to giving a brief overview of the modular objects relevant to our results. An excellent reference is Chapters 1 and 2 of [16]. We are interested in modular forms of integral or half-integral weight with multiplier system for the congruence subgroup

for an integer \(N \geqslant 1\). A multiplier system on \(\Gamma _{0}(N)\) is a function \(v : \Gamma _{0}(N) \rightarrow {\mathbb {C}}^{*}\) satisfying some consistency conditions. We will not need the details, so we refer the reader to [13, Sec. 2.6].

Let \({\mathfrak {H}}= \{ z \in {\mathbb {C}}\, | \, {\text {Im}}(z) > 0 \}\) be the upper-half plane in \({\mathbb {C}}\).

### Definition A.1

A holomorphic function \(f: {\mathfrak {H}}\rightarrow {\mathbb {C}}\) is called a *modular form* of weight \(k \in \mathbb {R}\) and multiplier system *v* on \(\Gamma _{0}(N)\) if *f* is holomorphic at all cusps \(\mathbb {Q}\cup \{\infty \}\), and if for all \(L \in \Gamma _{0}(N)\), *f* transforms as

We call *f* a *cusp form* if additionally, *f* vanishes at all cusps.

We employ the change of variables \(q = \exp (2 \pi i \tau )\) and with it, the abuse of notation writing \(f(\tau )\) and *f*(*q*) interchangeably. The fundamental building block of a large class of modular forms is the Dedekind eta function (or just eta function)

which is a modular form of weight \(\frac{1}{2}\) and multiplier system \(v_{\eta }\) on \({\text {SL}}_{2}({\mathbb {Z}})\); see [13, Sec. 2.8] where \(v_{\eta }\) is given explicitly.^{Footnote 9}

### Definition A.2

An *eta product* of level \(N \geqslant 1\) is a function \(f : {\mathfrak {H}}\rightarrow {\mathbb {C}}\) of the form

such that \(a_{m} \in {\mathbb {Z}}\) (possibly negative, or zero) for all *m*|*N*, and where the product is over positive divisors of *N*.

From the modular properties of the Dedekind eta function, one can show that an eta product *f* of level *N* transforms as a modular form on \(\Gamma _{0}(N)\) of weight

and with multiplier system

When we say “transforms as” we mean that *f* satisfies (26) for all \(L \in \Gamma _{0}(N)\). An eta product *f* is automatically holomorphic on \({\mathfrak {H}}\). This is because the form of \(\eta (q)\) indicates that any poles of *f* must occur at \(q=0\) or \(|q|=1\). All that is left to consider is when an eta product is holomorphic at the cusps. The following proposition gives necessary and sufficient conditions.

### Proposition A.3

( [16, Cor. 2.3]). An eta product *f* of level *N* is holomorphic at the cusps if and only if the following holds for all positive divisors *c* of *N*

Moreover, *f* vanishes at all cusps if and only if each inequality is strict. An eta product is therefore a modular form of weight *k* for \(\Gamma _{0}(N)\) if and only if each inequality is satisfied, and it is a cusp form if and only if each is strictly satisfied.

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Pietromonaco, S. *G*-invariant Hilbert schemes on Abelian surfaces and enumerative geometry of the orbifold Kummer surface.
*Res Math Sci* **9**, 1 (2022). https://doi.org/10.1007/s40687-021-00298-9

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DOI: https://doi.org/10.1007/s40687-021-00298-9