Which topologies are used by complex algebraic geometers in their usual expressions?

Question

A complex algebraicgeometer will encounter two topologies when studying a complex space. One is the analytic Zariski topology while the other one is the complex topology which is induced from its local model. You might take complex Euclidean space in your mind, then their open set are complement of zero set of holomorphic function and the usual open set (in Euclidean topology).

I want to known which topologies are used by complex algebraic geometers in their usual expressions? For example, in the famous book-Several complex variables VII, there are several propositions about open map.

(a) Every flat holomorphic map is open.

(b) Let $f: X \to Y$ be a holomorphic map of complex manifolds. If $f$ is open, then $f$ is flat.

In the above two Propositions, which topology is used to describe the openness of the map?

Answer 1

If you are talking about holomorphic maps, you want to look at the complex topology given by the charts of the manifold.

If the manifold happens to be a variety, we can also talk about the Zariski topology, but this doesn't coincide with the other topology and it doesn't make much sense to talk about holomorphic maps (regular functions would be the morphisms in the category of varieties). All of the theory of complex analysis is giving complex numbers their usual (Euclidean) topology so we can't really talk about holomorphic maps otherwise (you would lose a lot of nice properties and theorems!)

Answer 2

Question: "I want to known which topologies are used by complex algebraic geometers in their usual expressions?...In the above two Propositions, which topology is used to describe the openness of the map?"

Answer: You should state some definitions in the above mentioned book - there are readers on this "forum" that do not have a copy of the book.

Example: Let $k$ be the field of complex numbers, $X \subseteq T:=\mathbb{P}^n_k$ is a smooth and quasi projective algebraic variety and $U \subseteq X$ is a Zariski open set. Let $\mathcal{O}_X$ be the structure sheaf of $X$ in the sense of Hartshorne Chapter I.3. It follows any regular function $s\in \mathcal{O}_X(U)$ is holomorphic. A function on $U$ which locally is a rational function is holomorphic. If $\tau_Z$ is the Zariski open sets in $X$ and $\tau_s$ is the open subsets in the "strong" topology induced by the "strong" topology on $k$, 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 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. For any smooth quasi projective algebraic variety $X$ we may construct the corresponding "complex quasi projective manifold" $(X, \tau_s, \mathcal{O}_X^s)$ - here you must prove that $X$ has an atlas giving it the structure of a complex manifold, and that this structure is independent of choice of atlas. 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: This is expressed by saying there is an "equivalence of categories" between the category of coherent sheaves on $X$ viewed as algebraic variety and the category of coherent sheaves on $X$ viewed as a complex manifold - the above defined functor is an equivalence of categories - this is a part of the famous "GAGA" theorems from the 50s. Given any coherent $\mathcal{O}_X^s$-module $E^s$, there is a coherent $\mathcal{O}_X$-module $E$ and an isomorphism $F(E) \cong E^s$. There is moreover an equality

$$\eta: Hom_{\mathcal{O}_X^s}(F(E),F(G)) \cong Hom_{\mathcal{O}_X}(E,G).$$

For any map of coherent $\mathcal{O}_X$-modules $\phi: E \rightarrow G$, you get an induced map $F(\phi): F(E) \rightarrow F(G)$ inducing the bijection $\eta$.

You may define "flatness" of maps using the two structure sheaves $\mathcal{O}_X$ and $\mathcal{O}_X^s$. If you are doing algebraic geometry you use $\mathcal{O}_X$ and $\tau_Z$, if you are doing complex geometry you use $\mathcal{O}_X^s$ and $\tau_s$.

Note: The Zariski topology is the topology defined where the closed sets are zero sets of polynomials. I'm unsure what you mean when speaking of the "analytic Zariski topology".