What is the definition of the dimension of an algebraic manifold?

Question

I have a very basic question. It says on Wikipedia that an algebraic manifold is an algebraic variety which is also a manifold. So suppose I have an algebraic manifold $V$ which is an affine variety over $\mathbb{C}$ and also a manifold. Does one define the dimension of $V$ as an affine variety or as a manifold? or do the two notions always coincide? Thank you!

Answer 1

The two always coincide. One way to see this is to note that the completetion at a point of a smooth variety is isomorphic to $\mathbb{C}[[x_1, \dots x_n]]$, where $n$ is the algebraic dimension. This gives a morphism $\mathbb{C}[x_1, \dots x_n]\to \mathcal{O}_X$ by lifting elements, which gives a morphism $X\to \mathbb{A}^n$ that is an isomorphism on the tangent space. This implies the same condition on $X(\mathbb{C})\to \mathbb{C}^n$, and thus by the inverse function theorem, $X(\mathbb{C})$ is locally isomorphic to $\mathbb{C}^n$. This implies that the complex topological dimension of $X(\mathbb{C})$ concides with the algebraic dimension.