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Q:

Question about the definition of the “standard complex structure on $\mathbb{C}^n$”

I apologize for this question being so elementary, but I’m very new to complex analysis, and am looking for some assistance with the definition of the “standard complex structure on $\mathbb{C}^n$”.
If we fix some basis $e_1,\ldots,e_n$, then the standard complex structure on $\mathbb{C}^n$ is the one such that $J(e_1,\ldots,e_n)=(-1)^n(e_1,\ldots,e_n)$. And we say that we use a compatible complex structure when there is some linear map $J:\mathbb{C}^n\to\mathbb{C}^n$ such that $J(A\cdot B)=J(A)\cdot J(B)$ and $J^2=-I$ (the latter being the equivalent in Linear Algebra of the former).
So, my question is: is there an actual structure on $\mathbb{C}^n$ which comes with the standard complex structure (i.e., is it the one we get by restricting the structure of an algebra on $\mathbb{C}^n$ to $\mathbb{C}^n$)?
Thanks in advance!

A:

The only structure compatible with the standard complex structure in $\Bbb C^n$ is the product structure.

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