Let $L$ be a closed orientable Lagrangian embedding in $\mathbb C^n$. Then $\chi(L) = 0$, where $\chi(L)$ denotes the Euler characteristic of $L$.
This fact is more or less stated in section 3.2 of
and is attributed to Whitney. Does anyone know where can I find the original proof by Whitney? Thanks.
Remark: The proof is quite simple, This is because the self-intersection number of any submanifold of $\mathbb C^n$ is clearly zero, but it is also equal to the Euler characteristic of the normal bundle, which for Lagrangian submanifolds is isomorphic to the cotangent bundle. (Quoted from