Hamiltonian 2-forms in Kähler geometry, IV Weakly Bochner-flat Kähler manifolds
Bibliographic record
Abstract
We study the construction and classification of weakly Bochnerflat (WBF) metrics (i.e., Kähler metrics with coclosed Bochner tensor) on compact complex manifolds.A Kähler metric is WBF if and only if its 'normalized' Ricci form is a hamiltonian 2-form: such 2-forms were introduced and studied in previous papers in the series.It follows that WBF Kähler metrics are extremal.We construct many new examples of WBF metrics on projective bundles and obtain a classification of compact WBF Kähler 6-manifolds, extending work by the first three authors on weakly selfdual Kähler 4-manifolds.The constructions are independent of previous papers in the series, but the classification relies on the classification of compact Kähler manifolds with a hamiltonian 2-form [3]. 1 Introduction 92 2 Hamiltonian 2-forms and WBF Kähler metrics 94 2.1 Classification of hamiltonian 2-forms 94 2.2 Admissible bundles and metrics 95 2.3 WBF Kähler metrics of order 0 and 1 97 3 Kähler-Einstein metrics and Kähler-Ricci solitons 98 4 Constructions of WBF Kähler metrics 101 4.1 WBF Kähler metrics over a Kähler-Einstein manifold 103 91 92 Vestislav Apostolov et al. 4.2 WBF Kähler metrics over a product of Kähler-Einstein manifolds 107 4.3 WBF Kähler metrics over a ruled surface 111 4.4 WBF versus extremal Kähler metrics 113 5 Classification of WBF Kähler metrics on compact 6-manifolds 113 Appendix.Proofs of Lemmas 4.3, 4.4 and 4.5
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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 | 0.003 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| 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 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".