This problem was studied by Fujita in his paper ["Cancellation problem of complete varieties", Inventiones Mathematicae 64 (1981). ][1]

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):

>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 $V \cong W$.

In particular, cancellation problem has a positive answer for $M=\mathbb{P}^n$. 

The condition on the Albanese variety is a necessary one; in fact, it is known that cancellation is not always true for abelian varieties.

  [1]: http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN356556735_0064&DMDID=dmdlog13