{"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/on-the-cohomology-groups-of-real-lagrangians","title":"On the cohomology groups of real Lagrangians in Calabi-Yau threefolds","arxiv_id":"2002.03957","date":"2020-02-10","proceeding":null,"authors":["Hülya Argüz","Thomas Prince"],"abstract":"The quintic threefold $X$ is the most studied Calabi-Yau $3$-fold in the mathematics literature. In this paper, using \\v{C}ech-to-derived spectral sequences, we investigate the mod $2$ and integral cohomology groups of a real Lagrangian $\\breve{L}_{\\mathbb{R}}$, obtained as the fixed locus of an anti-symplectic involution in the mirror to $X$. We show that $\\breve{L}_{\\mathbb{R}}$ is the disjoint union of a $3$-sphere and a rational homology sphere. Analysing the mod $2$ cohomology further, we deduce a correspondence between the mod $2$ Betti numbers of $\\breve{L}_{\\mathbb{R}}$ and certain counts of integral points on the base of a singular torus fibration on $X$. By work of Batyrev, this identifies the mod $2$ Betti numbers of $\\breve{L}_{\\mathbb{R}}$ with certain Hodge numbers of $X$. Furthermore, we show that the integral cohomology groups $H^j(\\breve{L}_{\\mathbb{R}},\\mathbb{Z})$ of $\\breve{L}_{\\mathbb{R}}$ are $2$-primary for $j \\neq 0,3$; we conjecture that this holds in much greater generality.","url_abs":"https://arxiv.org/abs/2002.03957v2","url_pdf":"https://arxiv.org/pdf/2002.03957v2.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":"on-the-cohomology-groups-of-real-lagrangians","repo_url":"https://github.com/T-Prince/Cohomology-groups-of-real-Lagrangians-in-Calabi-Yau-threefolds","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null},{"paper_slug":"on-the-cohomology-groups-of-real-lagrangians","repo_url":"https://github.com/T-Prince/quintic-tropical-cycle","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}