Jim Bryan ; Ádám Gyenge - $G$-fixed Hilbert schemes on $K3$ surfaces, modular forms, and eta products

epiga:6986 - Épijournal de Géométrie Algébrique, March 9, 2022, Volume 6 - https://doi.org/10.46298/epiga.2022.6986
$G$-fixed Hilbert schemes on $K3$ surfaces, modular forms, and eta products

Authors: Jim Bryan ; Ádám Gyenge

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.

Volume: Volume 6
Published on: March 9, 2022
Accepted on: March 9, 2022
Submitted on: December 16, 2020
Keywords: Mathematics - Algebraic Geometry,14J28, 14C05, 11F03, 11F20, 11F27


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