Geometry and Analysis on Cauchy Riemann Manifolds John Bland (University of Toronto),
Bibliographic record
Abstract
One of the famous open questions in several complex variables is the following: Is every 5 dimensional strongly pseudoconvex CR manifold locally embeddable? This is a remarkably subtle question relating geometry, complex analysis and partial dierential equations. To shed some light on this question, we will brie\ny recall the denitions and known results. Let M be a smooth oriented 2n+1-manifold. An almost CR structure (for Cauchy-Riemann) on M is a complex subbundle H(1;0)M of the complexied tangent bundle such that H(0;1)M: = H(1;0)M is everywhere transverse to H(1;0)M. The almost CR structure is a CR structure if in addition H(0;1)M is integrable as a complex subbundle of the complexied tangent bundle TCM: Let be a real one-form annihiliating H(1;0)M; it is determined up to multiplication by a nonva-nishing function. We choose the sign of such that the orientation determined by and the natural orientation for H(1;0)M agrees with the orientation for M. We dene the Levi form associated to to be the Hermitian form on H(1;0)M determined by L(Z; W) = id(Z; W) for all Z; W 2 H(1;0)M. If the Levi form is positive denite, then the CR structure is said to be strongly pseudoconvex. This condition is independent of the choice of (except for its sign). Remark 1 Let U be a smoothly bounded strongly pseudoconvex open set in a complex manifold X. Then the complex structure from X restricts to @U as a strongly pseudoconvex CR structure; the Levi form dened here is the same as the usual Levi form dened in complex analysis. A CR function on M is a function f: M! C such that Z(f) = 0 for every Z 2 H(0;1)M. These are the tangential Cauchy-Riemann equations. We denote this @bf = 0. One of the most basic questions in the study of CR manifolds is the following: Question 2 Given a strongly pseudoconvex CR manifold (M; H(1;0)M), is it embeddable? That is, does there exist a smooth embedding X: M,! CN such the the components of X are CR functions:
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.002 | 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 teacher head, 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".