[2401.13056] Special metrics in hypercomplex geometry Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Differential Geometry arXiv:2401.13056 (math) [Submitted on 23 Jan 2024 ( v1 ), last revised 24 Apr 2026 (this version, v3)] Title: Special metrics in hypercomplex geometry Authors: Elia Fusi , Giovanni Gentili View a PDF of the paper titled Special metrics in hypercomplex geometry, by Elia Fusi and 1 other authors View PDF HTML (experimental) Abstract: We investigate the existence and geometric properties of special hyperhermitian metrics. First of all, we characterise hypercomplex structures with Obata holonomy in $\mathrm{SL}(n, \mathbb{H})$ in terms of the existence of quaternionic Gauduchon metrics together with the vanishing of a hypercomplex cohomological invariant. In view of this, the quaternionic Gauduchon and quaternionic balanced conditions are investigated at length: we describe their properties and determine criteria for their existence. Furthermore, we prove an incompatibility result concerning strong HKT and balanced hyperhermitian metrics, confirming an open conjecture by Fino and Vezzoni in the hypercomplex framework. Finally, we introduce an Einstein-type condition, determining basic properties, obstructions and providing examples. In particular, we show that Joyce's manifolds always admit such type of metrics. Comments: Treatment streamlined and condensed down to 34 pages. Final version to appear in Advances in Mathematics Subjects: Differential Geometry (math.DG) ; Complex Variables (math.CV) MSC classes: 53C26, 32W50, 53C25, 53C56 Cite as: arXiv:2401.13056 [math.DG] (or arXiv:2401.13056v3 [math.DG] for this version) https://doi.org/10.48550/arXiv.2401.13056 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Giovanni Gentili [ view email ] [v1] Tue, 23 Jan 2024 19:27:37 UTC (69 KB) [v2] Tue, 6 Feb 2024 21:48:08 UTC (52 KB) [v3] Fri, 24 Apr 2026 08:58:07 UTC (50 KB) Full-text links: Access Paper: View a PDF of the paper titled Special metrics in hypercomplex geometry, by Elia Fusi and 1 other authors 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.CV 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