Aug 2026· 1 citation· ⚡ 1 influential· 16 references
Mathematics
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 $$X\text{ is algebraic} \quad \Longleftrightarrow \quad \mathcal{H}^{2d} \bigl( \Sigma_\infty(X)\bigr)=0. $$ 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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Let $F$ and $E$ be finite extensions of $\mathbb Q_p$, let $\mathbb G$ be a reductive group over $F$, and put $G=\mathbb G(F)$. Let $V$ be a locally algebraic representation of the form $V=\pi_{\mathrm{sm}}\otimes_E\sigma_{\mathrm{alg}}$, where $\pi_{\mathrm{sm}}$ is smooth admissible and $\sigma_{\mathrm{alg}}$ is fin...
A. Pyvovarov· 0 citations
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