A mirror theorem for multi-root stacks and applications
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Bibliographic record
Abstract
Abstract Let X be a smooth projective variety with a simple normal crossing divisor $$D:=D_1+D_2+\cdots +D_n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>D</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>D</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>D</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:math> , where $$D_i\subset X$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>D</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>⊂</mml:mo> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> are smooth, irreducible and nef. We prove a mirror theorem for multi-root stacks $$X_{D,{\overrightarrow{r}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>X</mml:mi> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>,</mml:mo> <mml:mover> <mml:mi>r</mml:mi> <mml:mo>→</mml:mo> </mml:mover> </mml:mrow> </mml:msub> </mml:math> by constructing an I -function lying in a slice of Givental’s Lagrangian cone for Gromov–Witten theory of multi-root stacks. We provide three applications: (1) We show that some genus zero invariants of $$X_{D,\overrightarrow{r}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>X</mml:mi> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>,</mml:mo> <mml:mover> <mml:mi>r</mml:mi> <mml:mo>→</mml:mo> </mml:mover> </mml:mrow> </mml:msub> </mml:math> stabilize for sufficiently large $$\overrightarrow{r}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>r</mml:mi> <mml:mo>→</mml:mo> </mml:mover> </mml:math> . (2) We state a generalized local-log-orbifold principle conjecture and prove a version of it. (3) We show that regularized quantum periods of Fano varieties coincide with classical periods of the mirror Landau–Ginzburg potentials using orbifold invariants of $$X_{D,\overrightarrow{r}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>X</mml:mi> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>,</mml:mo> <mml:mover> <mml:mi>r</mml:mi> <mml:mo>→</mml:mo> </mml:mover> </mml:mrow> </mml:msub> </mml:math> .
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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 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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 it