On Hodge polynomials for nonalgebraic complex manifolds
Bibliographic record
Abstract
Hodge theory is pivotal in studying algebraic varieties' intricate geometry and topology: it provides essential insights into their structure. The Hodge decomposition theorem establishes a profound link between the geometry of varieties and their cohomology groups, helping to understand their underlying properties. Moreover, Hodge theory was crucial at the inception of the field of mirror symmetry, revealing deep connections among seemingly disparate algebraic varieties. It also sheds light on studying algebraic cycles and motives, crucial objects in algebraic geometry. This article explores Hodge polynomials and their properties, specifically focusing on non-Kähler complex manifolds. We investigate a diverse range of such manifolds, including (quasi-)Hopf, (quasi-)Calabi-Eckmann, and LVM manifolds, alongside a class of definable complex manifolds encompassing both algebraic varieties and the aforementioned special cases. Our research establishes the preservation of the motivic nature of Hodge polynomials inside this broader context. Through explicit calculations and thorough analyses, this work contributes to a deeper understanding of complex manifold geometry beyond the realm of algebraic varieties. The outcomes of this study have potential applications in various areas of mathematics and physics where complex manifolds play a significant role.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.000 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 0.001 |
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 source (direct Gemma or distilled Codex), 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".