Cohomology of quotients in real symplectic geometry
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Bibliographic record
Abstract
Let (, , , ) be a Hamiltonian system where (, ) is a compact connected symplectic manifold, is a compact connected Lie group acting symplectically on and : g * is a moment map where g = Lie().Fix an Ad-invariant inner product on g and consider the norm squared of the moment map = |||| 2 : R. Kirwan has proved that the function is -equivariantly perfect over the field of rational numbers and is -equivariantly formal.She also gives a recursive formula for the equivariant rational Betti numbers of the subspace 0 = -1 (0).Suppose that there exists a pair of involutions ( : , : ) in a Hamiltonian system such that is anti-symplectic and they are compatible in a way that the fixed point set of , denoted by , acts on the fixed point set of , denoted by .If : R is the restriction of to the real locus , we prove that under certain conditions, called 2-primitivity and free extension property, the restricted function is equivariantly perfect over the field Z 2 and the real locus is equivariantly formal over the field Z 2 .In particular, when = U() or SU() and the group involution is the complex conjugation, we compute the Z 2 -Betti numbers of the quotient space 0 / where 0 = -1 (0).iii A Complex Projective Line B Grassmannians
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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 | 0.002 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.004 | 0.006 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.008 | 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