Arithmetic normal functions and filtrations on Chow groups
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Bibliographic record
Abstract
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X slash double-struck upper C"> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">X/\mathbb {C}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a smooth projective variety, and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="CH Superscript r Baseline left-parenthesis upper X comma m right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>CH</mml:mtext> </mml:mrow> <mml:mi>r</mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>m</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\textrm {CH}^r(X,m)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the higher Chow group defined by Bloch. Saito and Asakura defined a descending candidate Bloch-Beilinson filtration <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="CH Superscript r Baseline left-parenthesis upper X comma m semicolon double-struck upper Q right-parenthesis equals upper F Superscript 0 Baseline superset-of midline-horizontal-ellipsis superset-of upper F Superscript r Baseline superset-of upper F Superscript r plus 1 Baseline equals upper F Superscript r plus 2 Baseline equals midline-horizontal-ellipsis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>CH</mml:mtext> </mml:mrow> <mml:mi>r</mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>m</mml:mi> <mml:mo>;</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>F</mml:mi> <mml:mn>0</mml:mn> </mml:msup> <mml:mo> ⊃ </mml:mo> <mml:mo> ⋯ </mml:mo> <mml:mo> ⊃ </mml:mo> <mml:msup> <mml:mi>F</mml:mi> <mml:mi>r</mml:mi> </mml:msup> <mml:mo> ⊃ </mml:mo> <mml:msup> <mml:mi>F</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>r</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>F</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>r</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>=</mml:mo> <mml:mo> ⋯ </mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\textrm {CH}^r(X,m;\mathbb {Q}) = F^0\supset \cdots \supset F^r\supset F^{r+1} = F^{r+2}=\cdots</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , using the language of mixed Hodge modules. Another more geometrically defined filtration is constructed by Kerr and Lewis in terms of germs of normal functions. We show that under the assumptions (i) <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X slash double-struck upper C equals upper X 0 times double-struck upper C"> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo> × </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">X/\mathbb {C} = X_0\times \mathbb {C}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X 0"> <mml:semantics> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">X_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is defined over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Q overbar"> <mml:semantics> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:mo accent="false"> ¯ </mml:mo> </mml:mover> <mml:annotation encoding="application/x-tex">\overline {\mathbb {Q}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and (ii) the general Hodge conjecture, that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F Superscript bullet Baseline CH Superscript r Baseline left-parenthesis upper X comma m semicolon double-struck upper Q right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>F</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> ∙ </mml:mo> </mml:mrow> </mml:msup>
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.
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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 it