TITLE: Variety vs. Manifold
QUESTION [14 upvotes]: 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.

REPLY [12 votes]: 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.