In the ambit of differential geometry the aim is to study smooth manifolds. Why the objects studied in algebraic geometry are called algebraic varieties and not for example algebraic manifolds? I am a fan of the uniqueness of mathematics, so I think that these differences of terminology or notation may mislead the student.
Variety vs. Manifold
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1In Spanish, *variedad* = manifold or variety… – 2014-07-26
2 Answers
Many algebraic varieties are not manifolds. For example, the coordinate axes in $\mathbb{R}^2$ are an algebraic variety, but not a manifold because it isn't locally homeomorphic to $\mathbb{R}$ at the origin.
Edit: Thank you to Robert for pointing out that the issue here isn't smoothness, I think I was slightly on autopilot. You can also get algebraic varieties which are non-smooth even though they are manifolds, such as this one. But as has been pointed out already, depending on your definition of manifold this may be fine anyway.
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0@Galoisfan I think I confused the issue by talking about smoothness, which isn't really the problem with my example anyway - it's just not a manifold at the origin, under any definition I know. – 2012-05-21
A variety does not qualify as a manifold for more reasons other than smoothness. For example the $xy$-plane union the $z$-axis is a variety. But, there isn't even a well-defined dimension there. You would need a sufficiently broad definition of manifold to include varieties that are not smooth and don't have a dimension. At that point, the word "manifold" would not be very useful.