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Let $X$ be a complex manifold, and let $O(X)$ be the ring of holomorphic functions on $X$ .

Is there any important relation between the locally ringed spaces $(\operatorname{Spec}( O(X)),O_{\operatorname{Spec}(O(X)})$ and $(X,O)$, where $O$ is the sheaf of holomorphic functions on U ?

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    Edit the title too.2012-04-16

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The Spec construction has the following property: there is a natural bijection $\textbf{LocRingSp}(X, \operatorname{Spec} A) \cong \textbf{CRing}(A, \mathscr{O}(X))$ Hence, there is always a morphism $X \to \operatorname{Spec} \mathscr{O}(X)$. Moreover, for any ring $B$ and any morphism $X \to \operatorname{Spec} B$, there is a unique factorisation through $\operatorname{Spec} \mathscr{O}(X)$. So you can think of $\operatorname{Spec} \mathscr{O}(X)$ as being the universal affine scheme which approximates $X$.

The case where $X$ is a connected complex projective manifold is somewhat uninteresting though: $\mathscr{O}(X)$ is just the ring of complex numbers $\mathbb{C}$, and $\operatorname{Spec} \mathscr{O}(X)$ is just a point. Perhaps you should be asking about relative Spec?