# Snazzy applications of Several Complex Variables techniques

I am starting to dive into a study of Several Complex Variables. I would like to have a few guiding examples of "big payoffs". These should be very natural sounding theorems which depend on a lot of the machinery of SCV to prove them in an efficient manner. Here are a few examples:

$1$. Let $U$ be an open set in $\mathbb{C}^n$. Say I have an $n\times m$ matrix $G$ of holomorphic functions on $U$ and an $n$ vector $W$ of holomorphic functions on $U$. Say that there is an open cover $U_i$ of $U$, and $m$ vectors of holomorphic functions $V_i$ defined on $U_i$ such that $G(V_i) = W$. Then there is a global solution of the equation $G(V) = W$ where $V$ is defined on all of $U$ whenever $U$ is a domain of holomorphy.

This theorem is certainly natural, easy to state, appealing, and useful. It is often easy to get local solutions to a system of equations, but deducing the existence of a global solution is hard. This theorem says that for a domain of holomorphy, local solutions always give rise to global ones.

$2$. Every Riemann surface has a globally defined meromorphic function.

Even though this looks like a single complex variable theorem, the proof I have seen places it naturally in the context of vanishing theorems for sheaf cohomology of complex analytic manifolds.

$3$. On a noncompact Riemann surface, every divisor is the divisor of a global meromorphic function, which is itself the ratio of two global holomorphic functions.

Again, this is really a single complex variable theorem, which seems to be easily dealt with using techniques generally developed in SCV. It generalizes the Weierstrass Theorem that there exist entire functions with prescribed zeros and poles with given orders. The proof uses information about Stein Manifolds, and some cohomological vanishing theorems.

I would like to see some more statements in the same vein, along with a brief description of the tool kits that they require. This way, as I am studying, I might be able to take a pause whenever I realize I have the required background, and prove some of these very natural theorems.

I apologize if the question isn't very clear. I just don't want to head blindly into the study of SCV - I might miss out on the reason all these things came to be studied in the first place.

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