A moduli curve for compact conformally-Einstein Kähler manifolds
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Bibliographic record
Abstract
We classify quadruples (M, g, m, ) in which (M, g) is a compact Khler manifold of complex dimension m > 2 and is a nonconstant function on M such that the conformally related metric g/ 2 , defined wherever = 0, is an Einstein metric. It turns out that M then is the total space of a holomorphic CP 1 bundle over a compact Khler-Einstein manifold (N, h). The quadruples in question constitute four disjoint families: one, well known, with Khler metrics g that are locally reducible; a second, discovered by Brard Bergery (1982), and having = 0 everywhere; a third one, related to the second by a form of analytic continuation, and analogous to some known Khler surface metrics; and a fourth family, present only in odd complex dimensions m 9. Our classification uses a moduli curve, which is a subset C, depending on m, of an algebraic curve in R 2 . A point (u, v) in C is naturally associated with any (M, g, m, ) having all of the above properties except for compactness of M , replaced by a weaker requirement of 'vertical' compactness. One may in turn reconstruct M , g and from (u, v) coupled with some other data, among them a Khler-Einstein base (N, h) for the CP 1 bundle M . The points (u, v) arising in this way from (M, g, m, ) with compact M form a countably infinite subset of C.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 0.006 |
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