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.
6$\begingroup$ I think there are counterexamples among Abelian varieties. $\endgroup$– Francesco PolizziNov 8, 2016 at 15:51
1$\begingroup$ Use tensor products of the associated algebras to look for couterexamples. $\endgroup$– Al-AmraniNov 8, 2016 at 16:25
$\begingroup$ Or just let $Y$ be empty ;-) $\endgroup$– zntNov 8, 2016 at 21:38
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.