$G$-fixed Hilbert schemes on $K3$ surfaces, modular forms, and eta products
Bibliographic record
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 $\Gamma_{0}(|G|)$. 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. Comment: Published version. Greatly simplified proof of Proposition 3.1
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.001 | 0.004 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.013 | 0.002 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".