Central Kähler metrics with non-constant central curvature
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Bibliographic record
Abstract
The <italic>central curvature</italic> of a Riemannian metric is the determinant of its Ricci endomorphism, while the scalar curvature is its trace. A Kähler metric is called <italic>central</italic> if the gradient of its central curvature is a holomorphic vector field. Such metrics may be viewed as analogs of the extremal Kähler metrics defined by Calabi. In this work, central metrics of non-constant central curvature are constructed on various ruled surfaces, most notably the first Hirzebruch surface. This is achieved via the momentum construction of Hwang and Singer, a variant of an ansatz employed by Calabi (1979) and by Koiso and Sakane (1986). Non-existence, real-analyticity and positivity properties of central metrics arising in this ansatz are also established.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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