Relative Riemann-Hilbert and Newlander-Nirenberg Theorems for torsion-free analytic sheaves on maximal and homogeneous spaces

[2506.05841] Relative Riemann-Hilbert and Newlander-Nirenberg Theorems for torsion-free analytic sheaves on maximal and homogeneous spaces Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Complex Variables arXiv:2506.05841 (math) [Submitted on 6 Jun 2025 ( v1 ), last revised 9 Apr 2026 (this version, v2)] Title: Relative Riemann-Hilbert and Newlander-Nirenberg Theorems for torsion-free analytic sheaves on maximal and homogeneous spaces Authors: Thomas Kurbach View a PDF of the paper titled Relative Riemann-Hilbert and Newlander-Nirenberg Theorems for torsion-free analytic sheaves on maximal and homogeneous spaces, by Thomas Kurbach View PDF HTML (experimental) Abstract: In this paper it is shown that for locally trivial complex analytic morphisms between some reduced spaces the Relative Riemann-Hilbert Theorem still holds up to torsion, i.e. tame flat relative connections on torsion-free sheaves are in 1-to-1 correspondence with torsion-free relative local systems. Subsequently, it is shown that generalised $\bar{\partial}$-operators on real analytic sheaves over complex analytic spaces can be viewed as relative complex analytic connections on the complexification of the underlying real analytic space with respect to a canonical morphism. By means of complexification, the Relative Riemann-Hilbert Theorem then yields a Newlander-Nirenberg type theorem for $\bar{\partial}$-operators on torsion-free real analytic sheaves over some complex analytic varieties. In the non-relative case, this result shows that on all maximal and homogeneous analytic spaces tame flat analytic connections are in 1-to-1 correspondence with local systems, which in turn are in 1-to-1 correspondence with linear representations of the fundamental group assuming connectedness. Comments: 68 pages, added acknowledgements Subjects: Complex Variables (math.CV) Cite as: arXiv:2506.05841 [math.CV] (or arXiv:2506.05841v2 [math.CV] for this version) https://doi.org/10.48550/arXiv.2506.05841 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Thomas Kurbach [ view email ] [v1] Fri, 6 Jun 2025 07:59:34 UTC (58 KB) [v2] Thu, 9 Apr 2026 10:15:05 UTC (58 KB) Full-text links: Access Paper: View a PDF of the paper titled Relative Riemann-Hilbert and Newlander-Nirenberg Theorems for torsion-free analytic sheaves on maximal and homogeneous spaces, by Thomas Kurbach View PDF HTML (experimental) TeX Source view license Current browse context: math.CV < prev | next > new | recent | 2025-06 Change to browse by: math 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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