Sorry for the naive question. Let $X\_1$, $X\_2$ and $Y$ be three projective varieties over an algebraically closed field of characteristic zero. If we have $X\_1\times Y\cong X\_2\times Y$, do we automatically get $X\_1\cong X\_2$? If not, do we have counter examples? If necessary, we could put stronger conditions (for example, smoothness) on the varieties.



Do we have "cancellation law" for products of varieties
You may look at T. Fujita's paper "Cancellation Problem of Complete Varieties". He proved that if $X\_1$ and $Y$ are "Picard independent" (look at Proposition 3 therein for a definition) then cancellation holds for any $X\_2$. In the same paper (Remark 8), the author cites T. Shioda "Some remarks on abelian varieties" in order to stress that cancellation does not hold for abelian varieties, as F. Polizzi said.


