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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_.2012-05-21
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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!2012-05-21
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    Maybe relevant: http://math.stackexchange.com/a/40154/32172012-05-21
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    Varieties tend to use the Zariski topology, which is wildly different from the topology of anything typically called a manifold.2012-05-21
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    Also maybe relevant: http://math.stackexchange.com/questions/9010/what-is-the-difference-between-a-variety-and-a-manifold2012-05-21
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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.2012-05-22
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    Also relevant is that there is no other term available than "variété" in French. I could come up with "multidim", "multiplat", "multidir", "multipart", "multigrandeur", "multicomposé(e)", "multicomp", or "pluri...". But the english language really has that power to accept such neologisms, to make them sound ok, that the french lacks. (Probably in part because english words are usually shorter and more flexible in pronunciation.)2012-05-22
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    It gets even worse in Russian, where both varieties and manifolds are called using one word "mnogoobrazie" :) The exact meaning can only be determined from the context.2012-05-26
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    In Spanish, *variedad* = manifold or variety…2014-07-26

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