Let $X \subseteq \mathbb{C}^n$ and suppose I can think of $X$ as a variety and as a manifold. Does the dimension as variety (I guess I just mean affine algebraic set) and as a manifold always coincide? Thank you very much!
Dimension of $X$ as a manifold and as a variety. Are they the same?
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algebraic-geometry
manifolds
1 Answers
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Yes, these two notions of dimension are the same. Let's assume that $X$ is irreducible; otherwise we can deal with each irreducible component separately. One way of seeing this is by using the Noether normalization theorem, which says that affine algebraic variety admits a finite map to $\mathbb{A}^n$, affine $n$-space. Using the going up and going down theorems, you can see that the algebraic dimension of $X$ will also be $n$. On the other hand, since $X$ has a finite map to $\mathbb{C}^n$, it will clearly have dimension $n$ as a manifold.