As a complex manifold $\mathbb{P}^n$ is locally the euclidean space $\mathbb{C}^n$, as a projective variety it is locally $\mathbb{C}^n$ with the Zariski topology, as a scheme it is locally $\text{Spec}(K[x_1,\ldots,x_n])$. It is well-known that the projective variety $\mathbb{P}^n$ is the set of the closed points of the scheme $\mathbb{P}^n=Proj(K[x_0,...,x_n])$. Is there any inspiring connection, e.g., between the goemetry of $\mathbb{P}^n$ as a scheme and the geometry of $\mathbb{P}^n$ as a complex manifold; more generally is there any inspiring connection between algebraic geometry(scheme theory) and complex manifolds?
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2$\begingroup$ A projective variety, even smooth, is, as a a rule, not locally $\mathbb C^n$ with Zariski topology. $\endgroup$– Serge LvovskiCommented Nov 25, 2015 at 14:45
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16$\begingroup$ The famous GAGA paper of Serre explains this connection. You cen get this paper here aif.cedram.org/aif-bin/item?id=AIF_1956__6__1_0 $\endgroup$– Liviu NicolaescuCommented Nov 25, 2015 at 14:48
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$\begingroup$ Existence (As finite étalé quotient of Hilbert Scheme) of moduli Space of Complex or Kähler Manifolds led to study of Scheme theory. For example, existence of moduli Space of log- Calabi Yau Manifolds still an open problem. $\endgroup$– user160903Commented Oct 15, 2020 at 16:38
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$\begingroup$ Study of "Douady space " play an important role in connection between Algebraic Geometry and Complex Geometry. $\endgroup$– user160903Commented Oct 15, 2020 at 16:42
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