G-invariant Hilbert schemes on Abelian surfaces and enumerative geometry of the orbifold Kummer surface

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

$$\begin{aligned} Z_{A, G}(q) = \sum _{d=0}^{\infty } e(\text {Hilb}^{d}(A)^{G})q^{d}. \end{aligned}$$

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.

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

$$\begin{aligned}\Gamma _{0}(N) := \bigg \{ \begin{pmatrix} a &{} b \\ c &{} d \end{pmatrix} \in {\text {SL}}_{2}({\mathbb {Z}}) \, \bigg | \, c \equiv 0 \ (\mathrm {mod}\ N) \bigg \} \subset {\text {SL}}_{2}({\mathbb {Z}}) \end{aligned}$$

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

$$\begin{aligned} f(L \tau ) = f\bigg (\frac{a \tau + b}{c \tau + d}\bigg ) = v(L) (c \tau +d)^{k} f(\tau ), \,\,\,\,\,\,\, L= \begin{pmatrix} a &{} b \\ c &{} d \end{pmatrix}. \end{aligned}$$

(26)

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)

$$\begin{aligned} \eta (q) = q^{\frac{1}{24}} \prod _{n=1}^{\infty }(1-q^{n}), \end{aligned}$$

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.⁹

Definition A.2

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

$$\begin{aligned} f(q) = \prod _{m|N} \eta (q^{m})^{a_{m}} \end{aligned}$$

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

$$\begin{aligned} k = \frac{1}{2} \sum _{m|N} a_{m} \in \tfrac{1}{2}{\mathbb {Z}}, \end{aligned}$$

and with multiplier system

$$\begin{aligned}v_{f}(L) = \prod _{m|N} \bigg (v_{\eta } \begin{pmatrix} a &{} mb \\ c/m &{} d \end{pmatrix}\bigg )^{a_{m}}, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, L= \begin{pmatrix} a &{} b \\ c &{} d \end{pmatrix}. \end{aligned}$$

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

$$\begin{aligned} \sum _{m|N} \frac{(\gcd (c,m))^{2})}{m} a_{m} \geqslant 0. \end{aligned}$$

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.