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I'm reading Hartshorne and in Chapter I Section 5, he says '... Over the complex numbers, for example, the nonsingular varieties are those which in the "usual" topology are complex manifolds'. I was wondering how to prove this using his definition of singular points. The nonsingular condition is quite reminiscent of the preimage theorem, so I'm thinking this could be relevant. Is there a reference for a proof of this fact?

Here is the definition of nonsingularily: Let $Y\subset \mathbb{A}^n$ be an affine variety, and let $f_1,\dots,f_t\in\mathbb{A}=k[x_1,\dots,x_n]$ be a set of generators for the ideal of $Y$. Y is nonsingular at a point $P\in Y$ if the rank of the matrix $\lVert\partial f_i/\partial x_j(P)\lVert$ is $n-r$, where $r$ is the dimension of $Y$. $Y$ is nonsingular if it is nonsingular at every point.

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Question: "I was wondering how to prove this using his definition of singular points. The nonsingular condition is quite reminiscent of the preimage theorem, so I'm thinking this could be relevant. Is there a reference for a proof of this fact?"

Answer: Let $k$ be the field of complex numbers, and let $(X,\mathcal{O}_X)$ be a smooth quasi projective algebraic variety over $k$ in the sense of Hartshorne, Chapter I. Let $U \subseteq X$ be a Zariski open set. It follows any regular function $s\in \mathcal{O}_X(U)$ is holomorphic. If $\tau_Z$ is the Zariski open sets in $X$ and $\tau_s$ is the open subsets in the "strong" topology, there is an inclusion $\tau_Z \subseteq \tau_s$ and a continuous map $id: (X, \tau_s) \rightarrow (X, \tau_Z)$: For any open set $U \in \tau_Z$ it follows $id^{-1}(U):=U \in \tau_s$. Hence the identity map is a continuous map. Let $\mathcal{O}_X^s$ denote the structure sheaf of holomorphic functions on $X$ in $\tau_s$. We get a map of ringed topological spaces

$$(id, id^{\#}):(X,\tau_s, \mathcal{O}_X^s) \rightarrow (X, \tau_Z, \mathcal{O}_X)$$

defined by sending a local sections $s \in \mathcal{O}_X(U)$ to "itself": We may view $s \in \mathcal{O}_X^s(U)$ since $\mathcal{O}_X(U) \subseteq \mathcal{O}_X^s(U)$ is a sub ring. Hence for any smooth quasi projective algebraic variety $X$ we may construct the corresponding "complex quasi projective manifold"

$$G(X, \mathcal{O}_X):=(X, \tau_s, \mathcal{O}_X^s).$$

It has the structure of a manifold because of the implicit function theorem and the fact that $X$ is non-singular. We may classify $X$ as algebraic variety and as complex manifold, and when doing this we use the structure sheaves $\mathcal{O}_X$ and $\mathcal{O}_X^s$. Changing the structure sheaf from the first to the second means changing from "rational functions" to "holomorphic functions". There is a canonical map

$$id^{\#}: id^{-1}(\mathcal{O}_X) \rightarrow \mathcal{O}_X^s$$

and for any coherent $\mathcal{O}_X$-module $\mathcal{E}$ we may construct

$$\mathcal{E}^s:=\mathcal{O}_X^s\otimes_{id^{-1}(\mathcal{O}_X)}id^{-1}(\mathcal{E}).$$

If $\mathcal{E}$ is locally free it follows $\mathcal{E}^s$ is locally free. We get a functor $F: Coh(\mathcal{O}_X) \rightarrow Coh(\mathcal{O}_X^s)$ defined by $F(\mathcal{E}):=\mathcal{E}^s$. When the variety $X$ is smooth and projective we do not "get anything new" when passing to holomorphic functions: The complex projective manifold $G(X, \mathcal{O}_X)$ is "uniquely determined" by $(X,\mathcal{O}_X)$.

When $X$ is projective there moreover an "equivalence of categories" between the category of coherent algebraic sheaves on $X$ viewed as algebraic variety and the category of coherent analytic sheaves on $X$ viewed as a complex manifold - the above defined functor is an equivalence of categories. Hence if you study vector bundles on complex projective manifolds you are studying algebraic varieties and algebraic geometry. You find a discussion with references in Hartshorne, Appendix B. The functor $F$ defined above is the functor studied in the GAGA papers from the 1950s.

Motivating (iso)morphism of varieties

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