# Birational invariants on cotangent bundles

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 $$Y$$.