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

TL;DR

This work analyzes the G-fixed Hilbert schemes on a complex K3 surface with a symplectic G-action by introducing the G-fixed partition function $Z_{X,G}(q)$ and proving that its reciprocal is a cusp form of weight $\tfrac12 e(X/G)$ for $\Gamma_0(|G|)$. It reduces the global problem to ADE-local contributions, giving an explicit eta-quotient formula $Z_{X,G}(q)=\eta^{-a}(k\tau)\prod Z_{\Delta_i}(k\tau/k_i)$, where the local factors are explicit: $Z_{A_n}(q)=1/\eta(\tau)$ and $Z_{\Delta}(q)$ for $D_n,E_n$ are eta-quotients tied to polyhedral symmetry data; a theta-function expression $Z_{\Delta}(q)=\theta_{\Delta}(\tau)/\eta(k\tau)^{n+1}$ provides new eta-product identities. The paper also extends these results to refinements such as the elliptic genus, the $\chi_y$ genus, and motivic classes, and proves a birational factorization for the corresponding partition function, linking Hilbert schemes on the quotient to those on a minimal resolution via delta-encoded theta-eta corrections. The 82 possible $(X,G)$ types are catalogued, and several arithmetic and geometric corollaries are drawn, including Hecke-eigenform phenomena in specific Xiao-number cases and connections to CHL models. Overall, the work blends deformation-theoretic, Nakajima-Quiver, and modular-form techniques to produce explicit eta-quotients and theta-eta identities governing fixed-point Hilbert schemes on K3 surfaces.

Abstract

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 $Γ_{0}(|G|)$. 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.

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Remark 1.5
• Theorem 1.6
• Corollary 1.7
• Remark 1.8
• Corollary 1.9
• Remark 1.10