Can anybody help me to prove the following result:

**Proposition**. Let $A$ and $B$ be abelian varieties over a field $k$ of characteristic zero. Assume that $A \times \mathbb{P}_k^n$ and $B \times \mathbb{P}_k^m$ are birational for some $n, m \geq 0$. Then $A$ and $B$ are isomorphic.

Is the result true in positive characteristic?