Polynomially convex embeddings of odd-dimensional closed manifolds
Bibliographic record
Abstract
Abstract It is shown that any smooth closed orientable manifold of dimension <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mrow><m:mn>2</m:mn><m:mo></m:mo><m:mi>k</m:mi></m:mrow><m:mo>+</m:mo><m:mn>1</m:mn></m:mrow></m:math> {2k+1} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>k</m:mi><m:mo>≥</m:mo><m:mn>2</m:mn></m:mrow></m:math> {k\geq 2} , admits a smooth polynomially convex embedding into <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msup><m:mi>ℂ</m:mi><m:mrow><m:mn>3</m:mn><m:mo></m:mo><m:mi>k</m:mi></m:mrow></m:msup></m:math> {\mathbb{C}^{3k}} . This improves by 1 the previously known lower bound of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mrow><m:mn>3</m:mn><m:mo></m:mo><m:mi>k</m:mi></m:mrow><m:mo>+</m:mo><m:mn>1</m:mn></m:mrow></m:math> {3k+1} on the possible ambient complex dimension for such embeddings (which is sharp when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>k</m:mi><m:mo>=</m:mo><m:mn>1</m:mn></m:mrow></m:math> {k=1} ). It is further shown that the embeddings produced have the property that all continuous functions on the image can be uniformly approximated by holomorphic polynomials. Lastly, the same technique is modified to construct embeddings whose images have nontrivial hulls containing no nontrivial analytic disks. The distinguishing feature of this dimensional setting is the appearance of nonisolated CR-singularities, which cannot be tackled using only local analytic methods (as done in earlier results of this kind), and a topological approach is required.
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How this classification was reachedexpand
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 | 0.004 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.004 | 0.003 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.005 |
| 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; both teacher heads agree on what is shown here.
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".