A Chow-Type theorem from asymptotic directions

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 $${\mathrm{\Sigma }}_{\mathrm{\infty }}(X):=\{\mathrm{\ell }\in {\mathbb{P}}^{n-1}(\mathbb{C}):\mathrm{\ell }\cap C_{\mathrm{\infty }}(X)\mathrm{\ne }\{0\}\},$$ where $C_{\mathrm{\infty }}(X)$ is the total tangent cone at infinity. We prove the metric Chow-type characterization $$X\text{ is algebraic}⟺{\mathcal{H}}^{2d}({\mathrm{\Sigma }}_{\mathrm{\infty }}(X))=0.$$ As an application, if $C_{\mathrm{\infty }}(X)$ is definable in an o-minimal expansion of $(\mathbb{R},+,\cdot )$ and ${\dim }_{\mathbb{R}}{C}_{\mathrm{\infty }}(X)\le 2d$, then $X$ is algebraic.