Euler characteristic is equal to self-intersection number of zero-section?
Solution 1:
Let's look at first definition. To compute the self-intersection number one needs to deform one of two copies of $\Delta$ slightly to make intersection transversal. It can be done in a small neighborhood of the diagonal — i.e. in the normal bundle of the diagonal in $X\times X$. But this normal bundle is isomorphic to the tangent bundle of $X$.