PomeLemma 22.1.96.AF0C82DE24E9 Let $X$ be a complex projective variety and $p:X\times \mathbb {C}\rightarrow X$ be the natural projection. Assume that $\mathcal{I}$ is an analytic coherent ideal sheaf on $X\times \mathbb {C}$. Assume that $\mathcal{I}|_{X \times \mathbb C^*}=p^*\mathcal{J}$ for some coherent ideal sheaf $\mathcal{J}$ on $X$. Then $\mathcal{I}$ is the analytification of an algebraic coherent ideal sheaf.
Lemma 22.1.96AF0C82DE24E9
Proof. Let $q:X\times (\mathbb {P}^1\setminus \{ 0\} )\rightarrow X$ be the natural projection. Since $\mathbb C^* \subset \mathbb {P}^1\setminus \{ 0\} $, we can glue $q^*\mathcal{J}$ with $\mathcal{I}$ to get an analytic coherent ideal sheaf on $X\times \mathbb {P}^1$. By the GAGA principle, this ideal sheaf is necessarily algebraic, hence so is its restriction to $X\times \mathbb {C}$. $\square$
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