Cohomology of fixed-point sets of anti-symplectic involutions of Hilbert schemes of points on a surface
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Bibliographic record
Abstract
Let S be a smooth, quasi-projective complex surface with complex symplectic form ω∈H0(S,KS). This determines a symplectic form ωn on the Hilbert scheme of points S[n] for n≥1. Let τ be an anti-symplectic involution of (S,ω): an order-two automorphism of S such that τ∗ω=−ω. Then τ induces an anti-symplectic involution on (S[n],ωn), and the fixed-point set (S[n])τ is a smooth Lagrangian subvariety of S[n]. In this paper, we calculate the mixed Hodge structure of H∗((S[n])τ;Q) in terms of the mixed Hodge structures of H∗(Sτ;Q) and of H∗(S∕τ;Q). We also classify the connected components of (S[n])τ and determine their mixed Hodge structures. Our results apply more generally whenever S is a smooth quasi-projective surface, and τ is an involution of S for which Sτ is a curve.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.004 | 0.007 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.004 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
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 it