Kähler metrics and toric Lagrangian fibrations

[2401.02910] Kähler metrics and toric Lagrangian fibrations Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Differential Geometry arXiv:2401.02910 (math) [Submitted on 5 Jan 2024 ( v1 ), last revised 14 Apr 2026 (this version, v2)] Title: Kähler metrics and toric Lagrangian fibrations Authors: Rui Loja Fernandes , Maarten Mol View a PDF of the paper titled K\"ahler metrics and toric Lagrangian fibrations, by Rui Loja Fernandes and Maarten Mol View PDF HTML (experimental) Abstract: We extend the Abreu-Guillemin theory of invariant Kähler metrics from toric symplectic manifolds to any symplectic manifold admitting a toric action of a symplectic torus bundle. We show that these are precisely the symplectic manifolds admitting a Lagrangian fibration with elliptic singularities. The base of such a toric Lagrangian fibration is a codimension 0 submanifold with corners of an integral affine manifold, called a Delzant subspace. This concept generalizes the Delzant polytope associated with a compact symplectic toric manifold. Given a Delzant subspace of finite type, we provide a Delzant-type construction of a Lagrangian fibration with the moment image being the specified Delzant subspace. We establish a 1:1 correspondence between invariant Kähler metrics and a pair consisting of an elliptic connection on the total space of the fibration and a hybrid $b$-metric on the base Delzant subspace, both with specified residues over the facets. Finally, we characterize extremal invariant Kähler metrics as those whose scalar curvature descends to an affine function on the base integral affine manifold. We show that this provides a method for finding and constructing extremal Kähler metrics. Comments: Final version accepted for publication in Journal of Symplectic Geometry Subjects: Differential Geometry (math.DG) ; Symplectic Geometry (math.SG) Cite as: arXiv:2401.02910 [math.DG] (or arXiv:2401.02910v2 [math.DG] for this version) https://doi.org/10.48550/arXiv.2401.02910 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Rui Loja Fernandes [ view email ] [v1] Fri, 5 Jan 2024 17:21:12 UTC (82 KB) [v2] Tue, 14 Apr 2026 20:24:28 UTC (75 KB) Full-text links: Access Paper: View a PDF of the paper titled K\"ahler metrics and toric Lagrangian fibrations, by Rui Loja Fernandes and Maarten Mol View PDF HTML (experimental) TeX Source view license Current browse context: math.DG < prev | next > new | recent | 2024-01 Change to browse by: math math.SG References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: 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) Major funding support from

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