Cohomology of fixed-point sets of anti-symplectic involutions of Hilbert schemes of points on a surface
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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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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 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".