Let $(\Sigma,g)$ be a closed Riemann surface with metric $g$. For any holomorphic line bundle $L\to \Sigma$, given a metric we have its curtature in terms of Chern connection $A$.
It is well-known that $\frac{i}{2\pi}\int_\Sigma F_A=\deg(L)$ is called the degree of Line bundle and $(\Sigma,g)$ induces a canonical orientation(as pointed by Yuan). $\frac{i}{2\pi} F_A\in H^2(\Sigma,\mathbb Z)$, the pairing with $[\Sigma]\in H_2(\Sigma,\mathbb Z)$ equals to $\deg(L)$.
Consider an examples: $\Sigma$ stays some oriented closed $4$-dimensional manfiold $M$, $L$ denotes the normal bundle.
Q
If we reverse the orientation of $M$, the intersection number will change, hence the degree will change. By the relation, $\deg(L)=[\Sigma]^2$, this intersection numner should be indepedent on the choice of orientation.
Of course somewhere must go wrong, could anyone give a help.