AI 中文总结
该研究针对闭纯$d$维复解析集，建立了基于射影渐近方向集的度量Chow型代数性刻画，并给出了无穷远切锥可定义且实维数不超过$2d$时复解析集为代数集的应用结论。
AI 中文摘要
设$X\subset \mathbb C^n$是闭的纯$d$维复解析集，我们为其关联射影渐近方向集$\Sigma_\infty(X):=\{\ell \in \mathbb P^{n-1}(\mathbb C):\ell\cap C_\infty(X)\ne\{0\}\}$，其中$C_\infty(X)$是无穷远的全切锥。我们证明了度量Chow型刻画：$X$是代数集当且仅当$\mathcal{H}^{2d}\bigl(\Sigma_\infty(X)\bigr)=0$。作为应用，若$C_\infty(X)$在$\mathbb{R}$的有序极小展开$(\\,+,\boldsymbol{\times}\\,)$中可定义，且$C_\infty(X)$的实维数$\dim_{\mathbb R} C_\infty(X)\le 2d$，则$X$是代数集。
英文摘要
Let $X\subset \mathbb C^n$ be a closed pure $d$-dimensional complex analytic set. We associate to $X$ its set of projective asymptotic directions $$ Σ_\infty(X) :=\{\ell \in \mathbb P^{n-1}(\mathbb C):\ell\cap C_\infty(X)\ne\{0\}\}, $$ where $ C_\infty(X)$ is the total tangent cone at infinity. We prove the metric Chow-type characterization $$X\text{ is algebraic} \quad \Longleftrightarrow \quad \mathcal{H}^{2d} \bigl( Σ_\infty(X)\bigr)=0. $$ As an application, if $C_\infty(X)$ is definable in an o-minimal expansion of $(\mathbb R,+,\cdot)$ and $\dim_{\mathbb R} C_\infty(X)\le 2d$, then $X$ is algebraic.
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