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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.

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Historical accident. Things are different in France, where smooth manifolds are variétés différentielles. –  Qiaochu Yuan May 21 '12 at 18:34
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IMO, the objects and techniques feel so different that it would be misleading to use the same term for both sorts! –  Hurkyl May 21 '12 at 18:37
    
Maybe relevant: math.stackexchange.com/a/40154/3217 –  Georges Elencwajg May 21 '12 at 19:13
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Varieties tend to use the Zariski topology, which is wildly different from the topology of anything typically called a manifold. –  MartianInvader May 21 '12 at 21:40
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@QiaochuYuan, nitpicking, differentiable manifolds are called as you say (or "variétés différentiables"), "smooth manifolds" is usually rather "variétés lisses". But you made a very good point that there is a historical component. Also the term "algebraic manifold" is highly used (and useful) to refer to algebraic varieties which are also manifolds. –  plm May 22 '12 at 11:33

2 Answers 2

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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The issue here is topology, not smoothness: the origin does not have a neighbourhood homeomorphic to $\mathbb R$. But there are other cases where smoothness is the issue, e.g. $y^2 - x^3 = 0$. –  Robert Israel May 21 '12 at 18:04
    
I agree with your answer, but one can say that some algebraic manifolds(the word manifold instead of the word variety) are not smooth manifolds. Why are used two different terms? It is more natural to fix the term manifold and to vary the adjective that could be smooth, algebraic, complex ecc. –  fair-coin tossing May 21 '12 at 18:05
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@Galoisfan: Sure, you could do that. But then you'd be speaking French and talking about variétés algébriques all the time. –  Zhen Lin May 21 '12 at 18:10
    
@ Zhen Lin: So, substantially the two terms have different origins; in particular the term variety comes from the french. Are there no mathematical motivations? (Infact manifold and variety are synonyms in enghish.) –  fair-coin tossing May 21 '12 at 18:14
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Une variété algébrique n'est pas une variété. –  Henry T. Horton May 21 '12 at 18:17

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.

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