Does V x C ~= C^{n+1} (isomorphism of varieties) imply that V ~= C^n?
This problem is as of yet unsolved, but many other things are not true: like in general, V x C ~= W x C does not imply V ~= W. My contribution consists of giving the ``best worst'' example yet, coming very close to the above cancellation problem.
One of the ingredients is Mason's ABC-theorem, whose counterpart in Number Theory is the notorious ABC-conjecture. (The first one is over C[X], the second one over Z.) The ABC conjecture, if true, trivially implies Fermat's Last Conjecture, and some similar ones, like ``x^a+y^b+z^c=0 + some requirements has no solutions''. The fact that these type of conjectures are all true for C[X] instead of Z, is what we use.