Positivity criteria for log canonical divisors and hyperbolicity
Bibliographic record
Abstract
Abstract Let X be a complex projective variety and D a reduced divisor on X . Under mild conditions on the singularities of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mo>(</m:mo><m:mi>X</m:mi><m:mo>,</m:mo><m:mi>D</m:mi><m:mo>)</m:mo></m:mrow></m:math> $(X,D)$ , which includes the case of smooth X with simple normal crossing D , and by running the minimal model program, we obtain by induction on dimension via adjunction geometric criteria guaranteeing various positivity conditions for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:msub><m:mi>K</m:mi><m:mi>X</m:mi></m:msub><m:mo>+</m:mo><m:mi>D</m:mi></m:mrow></m:math> $K_{X}+D$ . Our geometric criterion for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:msub><m:mi>K</m:mi><m:mi>X</m:mi></m:msub><m:mo>+</m:mo><m:mi>D</m:mi></m:mrow></m:math> $K_{X}+D$ to be numerically effective yields also a geometric version of the cone theorem and a criterion for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:msub><m:mi>K</m:mi><m:mi>X</m:mi></m:msub><m:mo>+</m:mo><m:mi>D</m:mi></m:mrow></m:math> $K_{X}+D$ to be pseudo-effective with mild hypothesis on D . We also obtain, assuming the abundance conjecture and the existence of rational curves on Calabi–Yau manifolds, an optimal geometric sharpening of the Nakai–Moishezon criterion for the ampleness of a divisor of the form <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:msub><m:mi>K</m:mi><m:mi>X</m:mi></m:msub><m:mo>+</m:mo><m:mi>D</m:mi></m:mrow></m:math> $K_{X}+D$ , a criterion verified under a canonical hyperbolicity assumption on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mo>(</m:mo><m:mi>X</m:mi><m:mo>,</m:mo><m:mi>D</m:mi><m:mo>)</m:mo></m:mrow></m:math> $(X,D)$ . Without these conjectures, we verify this ampleness criterion with mild assumptions on D , being none in dimension two and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>D</m:mi><m:mo>≠</m:mo><m:mn>0</m:mn></m:mrow></m:math> $D\neq 0$ in dimension three.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.005 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.001 | 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".