{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/enumerative-geometry-and-modularity-in-two","title":"Enumerative geometry and modularity in two-modulus K3-fibered Calabi-Yau threefolds","arxiv_id":"2408.02994","date":"2024-08-06","proceeding":null,"authors":["Charles Doran","Boris Pioline","Thorsten Schimannek"],"abstract":"Motivated in part by the modular properties of enumerative invariants of K3-fibered Calabi-Yau threefolds, we introduce a family of 39 Calabi-Yau mirror pairs $(X,Y)$ with $h_{1,1}(X)=h_{2,1}(Y)=2$, labelled by certain integer quadruples $(m,i,j,s)$ with $m\\leq 11$. On the A-model side, $X$ arises as a complete intersection in a projective bundle over a Fano fourfold $V_m^{[i,j]}$, and admits a Tyurin degeneration into a pair of degree $m$ Fano threefolds $F_m^{[i]}\\cup F_m^{[j]}$ intersecting on an anticanonical K3 divisor of degree $2m$. On the B-model side, $Y$ is fibered by $M_{m}$-polarized K3-surfaces of Picard rank 19, and determined by a branched covering of $\\mathbb{P}^1$, consistent with the Doran-Harder-Thompson mirror conjecture. When $s=0$, $Y$ itself acquires a Tyurin degeneration, and correspondingly $X$ acquires a fibration by degree $2m$ K3 surfaces, such that the two K\\\"ahler moduli control the size of the K3-fiber and base $\\mathbb{P}^1$. While the mirror pairs with $m\\leq 4$ can be realized as complete intersections in products of projective spaces or as hypersurfaces in toric varieties, the examples with $m\\geq 5$ are intrinsically non-toric. We obtain uniform formulae for the genus 0 and 1 topological free energies near the Tyurin degeneration (mirror to the large base limit), exhibiting modular properties under the Fricke-extended congruence group $\\Gamma_0(m)^+$. We use these results to compute the vertical Gopakumar-Vafa and Noether-Lefschetz invariants and check that their generating functions satisfy the expected modular properties. We also compute generating series of Gopakumar-Vafa invariants with fixed non-zero base degree and exhibit their modular properties.","url_abs":"https://arxiv.org/abs/2408.02994v1","url_pdf":"https://arxiv.org/pdf/2408.02994v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"enumerative-geometry-and-modularity-in-two","repo_url":"https://github.com/bpioline/twoparameterk3","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}