$V$, $W$ are varieties. Does $V\times \mathbf{P}^1=W\times \mathbf{P}^1$ imply $V=W$? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T16:40:22Z http://mathoverflow.net/feeds/question/78194 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/78194/v-w-are-varieties-does-v-times-mathbfp1w-times-mathbfp1-imply-v $V$, $W$ are varieties. Does $V\times \mathbf{P}^1=W\times \mathbf{P}^1$ imply $V=W$? lethe 2011-10-15T07:34:13Z 2011-10-15T23:30:14Z <p>If $\mathbf{P}^1$ is replaced by the affine line $\mathbf{A}^1$, this becomes the cancellation problem, and we have a pair of famous Danielewski surfaces ($xy=1-z^2$ and $x^2y=1-z^2$) as a counterexample (though I'm still seeking how to prove that..). I suppose in my case this counterexample might no longer work. </p> <p>Also one may replace $\mathbf{P}^1$ by $\mathbf{P}^n$ or other fixed varieties, or ask again after imposing some conditions on $V$ and $W$ (for example dimensions) if there are counterexamples for my question. And more wildly I may ask for what kind of family $X_n$, we will have the result that $V\times X_n=W\times X_n$ implies $V=W$. Any result of these kind of variations of the problem is also welcomed.</p> http://mathoverflow.net/questions/78194/v-w-are-varieties-does-v-times-mathbfp1w-times-mathbfp1-imply-v/78198#78198 Answer by Francesco Polizzi for $V$, $W$ are varieties. Does $V\times \mathbf{P}^1=W\times \mathbf{P}^1$ imply $V=W$? Francesco Polizzi 2011-10-15T09:01:58Z 2011-10-15T23:30:14Z <p>This problem was studied by Fujita in his paper <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN356556735_0064&amp;DMDID=dmdlog13" rel="nofollow">"Cancellation problem of complete varieties", Inventiones Mathematicae 64 (1981). </a></p> <p>He showed that the obstruction to cancellation is caused by the Picard schemes, proving the following remarkable result (see Corollary 7 in the cited paper):</p> <blockquote> <p>Let $M$, $V$ and $W$ be compact complex manifolds such that $M \times V \cong M \times W$. Assume that $M$ is projective and that $\textrm{Alb}(M)=0$ or $\textrm{Alb}(V)=0$. Then $V \cong W$.</p> </blockquote> <p>In particular, cancellation problem has a positive answer for $M=\mathbf{P}^n$. </p> <p>The condition on the Albanese variety is a necessary one; in fact, it is known that cancellation is not always true for abelian varieties.</p>