I think this question should have a well known answer, and I thanks in advance any help with it! Assume the base field is the complex numbers.
Question: Let $X$ and $Y$ be affine smooth irreducible varieties. If it is not the case that $X$ and $Y$ are birationally equivalent, is there any simple birational invariant that should distinguish the contangent bundles $T^*X$ and $T^*Y$, i. e., show that $T^*X$ and $T^*Y$ are also not birationally equivalent?
I have a special interest in the case of curves and surfaces, but a general answer would be nice! It would also be desirable that the birational invariant that distinguishes $T^*X$ from $T^*Y$ is related to the same invariant that distinguishes $X$ from $Y$.