度量几何Chow定理
Metric-geometric Chow theorem
AI总结：
本文推广o-minimal Chow定理，证明纯d维整复解析集为复代数集的5个等价度量几何条件，拓展了Chow定理的适用范围。
AI中文摘要：
2009年，Peterzil和Starchenko证明了Chow定理的如下优美推广：$\boldsymbol{\text{An entire complex analytic set }X\boldsymbol{\text{ in }}\boldsymbol{\text{C}}^n\boldsymbol{\text{ that is definable in an o-minimal structure on }}\boldsymbol{\text{R}}\boldsymbol{\text{ must be an algebraic set.}}$ 该结果如今被称为o-minimal Chow定理。本文中，我们给出了推广o-minimal Chow定理的Chow定理的若干几何与度量版本。例如，我们证明：若$X\boldsymbol{\text{是}}\boldsymbol{\text{C}}^n\boldsymbol{\text{中纯}}\boldsymbol{d}\boldsymbol{\text{维的整复解析集，则以下命题等价：(1) }X\boldsymbol{\text{是复代数集；(2) }}\boldsymbol{\text{H}}^{2d+1}(C(X,\boldsymbol{\text{∞}}))\boldsymbol{=0}$，其中$\boldsymbol{\text{H}}^k(A)\boldsymbol{\text{表示集合}}\boldsymbol{A}\boldsymbol{\text{的}}\boldsymbol{k}\boldsymbol{\text{维豪斯多夫测度，}}\boldsymbol{C(X,\boldsymbol{\text{∞}})}\boldsymbol{\text{表示}}\boldsymbol{X}\boldsymbol{\text{在无穷远处的切锥；(3) }}\boldsymbol{\text{H}}^{2d+2}(C_{\boldsymbol{\text{C}}}(X,\boldsymbol{\text{∞}}))\boldsymbol{=0}$，其中$\boldsymbol{C}_{\boldsymbol{\text{C}}}(X,\boldsymbol{\text{∞}})\boldsymbol{\text{表示}}\boldsymbol{X}\boldsymbol{\text{在无穷远处的复切锥；(4) }}\boldsymbol{\text{H}}^{2d}(Z(X,\boldsymbol{\text{∞}}))\boldsymbol{=0}$，其中对任意$\boldsymbol{A}\boldsymbol{\text{在}}\boldsymbol{\text{C}}^k\boldsymbol{\text{中，}}\boldsymbol{Z(A,\boldsymbol{\text{∞}})}\boldsymbol{=\boldsymbol{\text{[v]}\boldsymbol{\text{在}}\boldsymbol{\text{CP}}^{k-1}\boldsymbol{\text{中；}}\boldsymbol{v}\boldsymbol{\text{在}}\boldsymbol{C}_{\boldsymbol{\text{C}}}(A,\boldsymbol{\text{∞}})\boldsymbol{\text{中}}}$；(5) 对任意$\boldsymbol{k}\boldsymbol{\text{在}}\boldsymbol{\text{\textlbrace d+1,...,n\textrbrace}}\boldsymbol{\text{中，以及任意投影}}\boldsymbol{\boldsymbol{\text{π：}}\boldsymbol{\text{C}}^n\boldsymbol{\text{→}}\boldsymbol{\text{C}}^k}\boldsymbol{\text{满足}}\boldsymbol{\boldsymbol{\text{π}}^{-1}(0)\boldsymbol{\text{∩}}\boldsymbol{C}_{\boldsymbol{\text{C}}}(X,\boldsymbol{\text{∞}})\boldsymbol{=\boldsymbol{\text{\textlbrace 0\textrbrace}}}\boldsymbol{\text{，且}}\boldsymbol{Y\boldsymbol{=\boldsymbol{\text{π}}(X)}}\boldsymbol{\text{，则}}\boldsymbol{\text{H}}^{2d}(Z(Y,\boldsymbol{\text{∞}}))\boldsymbol{<\boldsymbol{\text{H}}^{2d}(\boldsymbol{\text{CP}}^d)}$。
英文摘要：
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 $π\colon \mathbb{C}^{n}\to \mathbb{C}^{k}$ such that $π^{-1}(0)\cap C_{\mathbb{C}}(X,\infty)=\{0\}$ and $Y=π(X)$, $\mathcal{H}^{2d}(Z(Y,\infty))<\mathcal{H}^{2d}(\mathbb{C}P^{d})$.
