A Chow-Type theorem from asymptotic directions

Abstract

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 $${\Sigma }_{\infty }(X):=\{\ell \in {\mathbb{P}}^{n-1}(\mathbb{C}):\ell \cap C_{\infty }(X)\ne \{0\}\},$$ where $C_{\infty }(X)$ is the total tangent cone at infinity. We prove the metric Chow-type characterization $$X\text{ is algebraic}⟺{\mathcal{H}}^{2d}({\Sigma }_{\infty }(X))=0.$$ 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.

9 pages