Weighted K–stability and coercivity withapplications to extremal Kähler and Sasaki metrics
Bibliographic record
Abstract
We show that a compact weighted extremal Kähler manifold, as defined by the third author (2019), has coercive weighted Mabuchi energy with respect to a maximal complex torus T C in the reduced group of complex automorphisms.This provides a vast extension and a unification of a number of results concerning Kähler metrics satisfying special curvature conditions, including Kähler metrics with constant scalar curvature, extremal Kähler metrics, Kähler-Ricci solitons, and their weighted extensions.Our result implies the strict positivity of the weighted Donaldson-Futaki invariant of any nonproduct T C -equivariant smooth Kähler test configuration with reduced central fibre, a property known as T C -equivariant weighted K-polystability on such test configurations.It also yields the T C -uniform weighted K-stability on the class of smooth T C -equivariant polarized test configurations with reduced central fibre.For a class of fibrations constructed from principal torus bundles over a product of Hodge cscK manifolds, we use our results in conjunction with results of Chen and Cheng (2021), He (2019) and Han and Li (2022) in order to characterize the existence of extremal Kähler metrics and Calabi-Yau cones associated to the total space, in terms of the coercivity of the weighted Mabuchi energy of the fibre.This yields a new existence result for Sasaki-Einstein metrics on certain Fano toric fibrations, extending the results of Futaki, Ono and Wang (2009) in the toric Fano case, and of Mabuchi and Nakagawa (2013) in the case of Fano P 1 -bundles.
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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 | 0.002 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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".