A moduli curve for compact conformally-Einstein Kähler manifolds
Bibliographic record
Abstract
We classify quadruples $(M, g, m, \tau)$ in which ( M , g ) is a compact Kähler manifold of complex dimension m > 2 and $\tau$ is a nonconstant function on M such that the conformally related metric $g/\tau^{2}$ , defined wherever $\tau \ne 0$ , is an Einstein metric. It turns out that M then is the total space of a holomorphic $\mathbb{C}{\rm P}^1$ bundle over a compact Kähler–Einstein manifold ( N , h ). The quadruples in question constitute four disjoint families: one, well known, with Kähler metrics g that are locally reducible; a second, discovered by Bérard Bergery (1982), and having $\tau \ne 0$ everywhere; a third one, related to the second by a form of analytic continuation, and analogous to some known Kähler surface metrics; and a fourth family, present only in odd complex dimensions $m \ge 9$ . Our classification uses a moduli curve, which is a subset $\mathcal{C}$ , depending on m , of an algebraic curve in $\mathbb{R}^2$ . A point ( u , v ) in $\mathcal{C}$ is naturally associated with any $(M, g, m, \tau)$ 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 $\tau$ from ( u , v ) coupled with some other data, among them a Kähler–Einstein base ( N , h ) for the $\mathbb{C}{\rm P}^1$ bundle M . The points ( u , v ) arising in this way from $(M, g, m, \tau)$ with compact M form a countably infinite subset of \mathcal{C}$ .
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.005 | 0.003 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.003 | 0.003 |
| Open science | 0.000 | 0.002 |
| Research integrity | 0.001 | 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".