Let $X$ and $Y$ be varieties with ample canonical sheaf. (We work over the complex numbers and varieties are smooth projective and connected.)
Suppose that $X$ and $Y$ are birationally equivalent. Are $X$ and $Y$ isomorphic?
Let $d$ be the dimension of $X$.
If $d=1$, the answer is yes. (Curves are determined by their function field.)
If $d=2$, the answer is yes. (The minimal model of a surface exists and is unique.)