When does variété mean manifold?

Solution 1:

Nowadays (note the caveat!) the translation into English of "variété algébrique" is definitely "algebraic variety". To give an explicit example, the equation $y^2=x^2+x^3$ describes "une variété algébrique affine" and in English "an affine algebraic variety" , not "an algebraic manifold". Traditionally, if you wanted to translate "algebraic manifold" into French you would say "une variété algébrique non-singulière": the adjectives "régulière" is more recent and "lisse" even more recent (due to Grothendieck I would guess). I even remember reading the clumsy sentence "une variété algébrique non nécessairement non-singulière" just meaning "an algebraic variety" !

A standard pun/joke is "une variété algébrique n'est pas une variété" which translated becomes "an algebraic variety is not a manifold", a true but not sidesplitting statement.

Finally, in (differential ) topology I have never heard anything but variété suitably qualified: topologique, différentiable (différentielle), $\mathcal C^k$, à bord. The translations would be : topological, differentiable (differential), $\mathcal C^k$, manifold and manifold with boundary .