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PreprintCC BY 4.0arXiv

Metric-geometric Chow theorem

  • José Edson Sampaio
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Abstract

In 2009, Peterzil and Starchenko proved the following beautiful generalization of Chow's theorem: An entire complex analytic set X⊂ ℂn that is definable in an o-minimal structure on 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⊂ ℂn is a pure d-dimensional entire complex analytic set, then the following statements are equivalent: (1) X is a complex algebraic set; (2) ℋ2d+1(C(X,∞))=0, where ℋk(A) denotes the k-dimensional Hausdorff measure of A, and C(X,∞) denotes the tangent cone at infinity of X; (3) ℋ2d+2(Cℂ(X,∞))=0, where Cℂ(X,∞) denotes the complex tangent cone at infinity of X; (4) ℋ2d(Z(X,∞))=0, where for A⊂ ℂk, Z(A,∞)={[v]∈ ℂPk−1;v∈ Cℂ(A,∞)}; (5) For any k∈{d+1,...,n} and for any projection π: ℂn→ ℂk such that π−1(0)∩ Cℂ(X,∞)={0} and Y=π(X), ℋ2d(Z(Y,∞))<ℋ2d(ℂPd).

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Cite

BibTeXRIS
BibTeX
@misc{sampaio2026metricgeometric,
  title = {{Metric-geometric Chow theorem}},
  author = {José Edson Sampaio},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.27810},
  eprint = {2608.27810},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.27810},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Metric-geometric Chow theorem
AU  - José Edson Sampaio
PY  - 2026
DA  - 2026/08/28
PB  - arXiv
DO  - 10.48550/arxiv.2608.27810
UR  - https://arxiv.org/abs/2608.27810
AB  - In 2009, Peterzil and Starchenko proved the following beautiful generalization of Chow's theorem: An entire complex analytic set X⊂ ℂ^n that is definable in an o-minimal structure on 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⊂ 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) ℋ^2d+1(C(X,∞))=0, where ℋ^k(A) denotes the k-dimensional Hausdorff measure of A, and C(X,∞) denotes the tangent cone at infinity of X; (3) ℋ^2d+2(C_ℂ(X,∞))=0, where C_ℂ(X,∞) denotes the complex tangent cone at infinity of X; (4) ℋ^2d(Z(X,∞))=0, where for A⊂ ℂ^k, Z(A,∞)={[v]∈ ℂP^k−1;v∈ C_ℂ(A,∞)}; (5) For any k∈{d+1,...,n} and for any projection π: ℂ^n→ ℂ^k such that π^−1(0)∩ C_ℂ(X,∞)={0} and Y=π(X), ℋ^2d(Z(Y,∞))<ℋ^2d(ℂP^d).
ER  -