부산대학교 도서관

If $X$ is a complex $K3$ surface with an effective action of a finite group $G$ which preserves the holomorphic symplectic form, then the possible $G$ are precisely the subgroups of the Mathieu group $M_{23}\subset M_{24}$ such that the induced action on the set $\{1,\dots,24\}$ has at least five orbits (Mukai) and all possible actions are classified into 82 possible topological types of the quotient $X/G$ (Xiao). \par Let $\Bbb{H}$ be the upper half-plane, $\tau\in\Bbb{H}$ and $q=\exp(2\pi i\tau)$. The $G$-fixed partition function of $X$ is defined by $$ Z_{X,G}(q)=\sum_{n=0}^{\infty}e({\rm Hilb}^n(X)^G)q^{n-1}, $$ where ${\rm Hilb}^n(X)^G$ is the $G$-fixed Hilbert scheme (or the $G$-equivariant Hilbert scheme or the $G$-invariant Hilbert scheme) of $X$ that parameterizes $G$-invariant length $n$ subschemes $Z\subset X$, and $e(-)$ is the topological Euler characteristic. \par The main result of this paper is as follows: \par Theorem 1.1. The function $Z_{X,G}(q)^{-1}$ is a modular cusp form of weight $\frac{1}{2}e(X/G)$ for the congruence subgroup $\Gamma_0(|G|)$. \par The authors obtain several formulas including an explicit formula for $Z_{X,G}$ in terms of the Dedekind eta function for all 82 possible $(X,G)$ and extend the results to various refinements of the Euler characteristic. They also apply their method to obtain an eta product identity for a certain shifted theta function of the root lattice of a simply laced root system. \par To prove their results, the authors use the fact that the Hilbert schemes of the stack $\Cal{X}=[\Bbb{C}^2/\{\pm 1\}]$ and the Hilbert schemes of the space $Y={\rm Tot}(K_{\Bbb{P}^1})$ can both be realized as moduli spaces of quiver representations of the $A_1$ Nakajima quiver variety and the work of Nakajima: the partition function of the Euler characteristics of the Hilbert scheme of points on the stack quotient $[\Bbb{C}^2/G_\Delta]$ can be computed explicitly in terms of the root data of $\Delta$.