$G$-fixed Hilbert schemes on $K3$ surfaces, modular forms, and eta products
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arXiv
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| Format: | Preprint |
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2019
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| _version_ | 1866915252544733184 |
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| author | Bryan, Jim Gyenge, Ádám |
| author_facet | Bryan, Jim Gyenge, Ádám |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1907_01535 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | $G$-fixed Hilbert schemes on $K3$ surfaces, modular forms, and eta products Bryan, Jim Gyenge, Ádám Algebraic Geometry 14J28, 14C05, 11F03, 11F20, 11F27 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. |
| title | $G$-fixed Hilbert schemes on $K3$ surfaces, modular forms, and eta products |
| topic | Algebraic Geometry 14J28, 14C05, 11F03, 11F20, 11F27 |
| url | https://arxiv.org/abs/1907.01535 |