Metric-geometric Chow theorem

[2608.27810] Metric-geometric Chow theorem Skip to main content Search Submit Donate Log in Mathematics > Algebraic Geometry arXiv:2608.27810 (math) [Submitted on 28 Aug 2026] Title: Metric-geometric Chow theorem Authors: José Edson Sampaio View PDF HTML (experimental) Abstract: In 2009, Peterzil and Starchenko proved the following beautiful generalization of Chow's theorem: An entire complex analytic set $X\subset \mathbb{C}^n$ that is definable in an o-minimal structure on $\mathbb{R}$ must be an algebraic set. This result is known nowadays as the o-minimal Chow theorem. In this article, we present some geometric and metric versions of Chow's theorem that generalize the o-minimal Chow's theorem. For instance, we prove that if $X\subset \mathbb{C}^n$ is a pure $d$-dimensional entire complex analytic set, then the following statements are equivalent: (1) $X$ is a complex algebraic set; (2) $\mathcal{H}^{2d+1}(C(X,\infty))=0$, where $\mathcal{H}^{k}(A)$ denotes the $k$-dimensional Hausdorff measure of $A$, and $C(X,\infty)$ denotes the tangent cone at infinity of $X$; (3) $\mathcal{H}^{2d+2}(C_{\mathbb{C}}(X,\infty))=0$, where $C_{\mathbb{C}}(X,\infty)$ denotes the complex tangent cone at infinity of $X$; (4) $\mathcal{H}^{2d}(Z(X,\infty))=0$, where for $A\subset \mathbb{C}^k$, $Z(A,\infty)=\{[v]\in \mathbb{C}P^{k-1};v\in C_{\mathbb{C}}(A,\infty)\}$; (5) For any $k\in\{d+1,...,n\}$ and for any projection $\pi\colon \mathbb{C}^{n}\to \mathbb{C}^{k}$ such that $\pi^{-1}(0)\cap C_{\mathbb{C}}(X,\infty)=\{0\}$ and $Y=\pi(X)$, $\mathcal{H}^{2d}(Z(Y,\infty))<\mathcal{H}^{2d}(\mathbb{C}P^{d})$. Comments: This paper appeared firstly on ResearchGate. 10 pages Subjects: Algebraic Geometry (math.AG) ; Complex Variables (math.CV) MSC classes: 32S20, 14B05, 32A15, 32S50 Cite as: arXiv:2608.27810 [math.AG] (or arXiv:2608.27810v1 [math.AG] for this version) https://doi.org/10.48550/arXiv.2608.27810 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Jose Edson Sampaio [ view email ] [v1] Fri, 28 Aug 2026 00:52:13 UTC (8 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.AG < prev | next > new | recent | 2026-08 Change to browse by: math math.CV 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) Major funding support from (opens in new tab) (opens in new tab) (opens in new tab)

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