Multi-valued Morse homotopy for the SYZ mirror of the complex projective plane

[2605.23404] Multi-valued Morse homotopy for the SYZ mirror of the complex projective plane Skip to main content Search Submit Donate Log in Mathematics > Symplectic Geometry arXiv:2605.23404 (math) [Submitted on 22 May 2026 ( v1 ), last revised 25 Aug 2026 (this version, v2)] Title: Multi-valued Morse homotopy for the SYZ mirror of the complex projective plane Authors: Hayato Nakanishi , Yat-Hin Suen View PDF HTML (experimental) Abstract: We propose a definition of the category $\mathit{Mo}^{\mathrm{mult}}(P)$ of multi-valued Morse homotopy on $P$ consisting of multi-valued functions associated to Lagrangian multi-sections. We then show that a full subcategory $\mathit{Mo}^{\mathrm{mult}}_{\mathcal{E}}(P)$ of $\mathit{Mo}^{\mathrm{mult}}(P)$ is $A_\infty$-equivalent to a full subcategory $\mathit{DG}_{\mathcal{E}}^{\mathrm{vect}}(\mathbb{C}P^2)$ of the category $\mathit{DG}^{\mathrm{vect}}(\mathbb{C}P^2)$ consisting of holomorphic vector bundles over the complex projective plane $\mathbb{C}P^2$. As an application, we study the mirror description for global sections of the holomorphic tangent bundle over $\mathbb{C}P^2$. Comments: 37 pages, 23 figures. Several typos are fixed, and new sections are added with examples of non-trivial jagged gradient trees to explain the necessity of the jagged gradient tree. Comments are welcome! Subjects: Symplectic Geometry (math.SG) ; Algebraic Geometry (math.AG); Differential Geometry (math.DG) Cite as: arXiv:2605.23404 [math.SG] (or arXiv:2605.23404v2 [math.SG] for this version) https://doi.org/10.48550/arXiv.2605.23404 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Hayato Nakanishi [ view email ] [v1] Fri, 22 May 2026 09:15:01 UTC (260 KB) [v2] Tue, 25 Aug 2026 05:34:30 UTC (310 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.SG < prev | next > new | recent | 2026-05 Change to browse by: math math.AG math.DG References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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