A twistor transform and normal forms for Cauchy Riemann structures
Bibliographic record
Abstract
Abstract We use Hitchin’s twistor transform for two-dimensional projective structures to obtain normal coordinates in a pseudoconcave neighbourhood of an <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">O</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \mathcal{O}(1) rational curve; in the construction, we present every such neighbourhood as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>Q</m:mi> <m:mi mathvariant="double-struck">D</m:mi> </m:msub> <m:mo>/</m:mo> <m:mi mathvariant="script">F</m:mi> </m:mrow> </m:math> Q_{\mathbb{D}}/\mathcal{F} for some holomorphic foliation ℱ, where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>Q</m:mi> <m:mi mathvariant="double-struck">D</m:mi> </m:msub> </m:math> Q_{\mathbb{D}} is an open neighbourhood in the standard quadric <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Q</m:mi> <m:mo>⊂</m:mo> <m:mrow> <m:msup> <m:mi mathvariant="double-struck">P</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo lspace="0.222em" rspace="0.222em">×</m:mo> <m:msup> <m:mi mathvariant="double-struck">P</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:mrow> </m:math> Q\subset\mathbb{P}^{2}\times\mathbb{P}^{2} . As a consequence of the normal coordinates, we obtain a new normal form for Cauchy Riemann structures on the three-sphere that are isotopic to the standard one. We end the paper with explicit calculations for the cases arising from deformations of the normal isolated singularities <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>X</m:mi> <m:mo></m:mo> <m:mi>Y</m:mi> </m:mrow> <m:mo>=</m:mo> <m:msup> <m:mi>Z</m:mi> <m:mi>n</m:mi> </m:msup> </m:mrow> </m:math> XY=Z^{n} .
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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 | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 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".