Weighted K-stability of polarized varieties and extremality of Sasaki\n manifolds
Bibliographic record
Abstract
We use the correspondence between extremal Sasaki structures and weighted\nextremal Kahler metrics defined on a regular quotient of a Sasaki manifold,\nestablished by the first two authors, and Lahdili's theory of weighted\nK-stability in order to define a suitable notion of (relative) weighted\nK-stability for compact Sasaki manifolds of regular type. We show that the\n(relative) weighted K-stability with respect to a maximal torus is a necessary\ncondition for the existence of a (possibly irregular) extremal Sasaki metric.\nWe also compare weighted K-stability to the K-stability of the corresponding\npolarized affine cone (introduced by Collins-Szekelyhidi), and prove that they\nagree on the class of test configurations we consider. As a byproduct, we\nstrengthen the obstruction to the existence of a scalar-flat Kahler cone metric\nfrom the K-semistability to the K-stability on these test configurations. We\nuse our approach to give a characterization of the existence of a compatible\nextremal Sasaki structure on a principal circle bundle over an admissible ruled\nmanifold, expressed in terms of the positivity of a single polynomial of one\nvariable over a given interval.\n
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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) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.001 |
| 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 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".