When learning about Riemann surfaces we encounter the fact that an analytic function germ on a small complex disk yields a Riemann surface via analytic continuation. It seems to me that the same construction should work in higher dimensions, to produce a complex manifold from a function germ. Is that any place in the literature where this is used and some examples are provided?
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asked Jan 15, 2022 at 11:40
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1$\begingroup$ The espace etale of the structure sheaf, with its connected components, exists for every locally ringed space. $\endgroup$– Jason StarrCommented Jan 15, 2022 at 12:19
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1$\begingroup$ The construct is the same, if $f$ is analytic at $z_0$ then consider each curve $\gamma:z_0\to z$ such that $f$ extends analytically along $\gamma$, call $f_\gamma$ the obtained analytic function around $z$, then $\{(z,f_\gamma)\}$ is a complex manifold. I heard that a theorem on Stein manifolds says that somehow you can replace $f_\gamma$ by finitely many partial derivatives $\partial^{\alpha_1}f_\gamma(z),\ldots,\partial^{\alpha_N}f_\gamma(z)$ maybe giving a global embedding into $\Bbb{C}^{n+N}$. $\endgroup$– reunsCommented Jan 15, 2022 at 12:46
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