Let $C$ a complex curve and $L$ a holomorphic line bundle on it.
I want to show that $L$ is positive iff it has positive degree.
Here the degree is defined as $\int_C c_1(L)$ and positive means that $c_1(L)$ can be represented by a positive closed real (1,1)-form, i.e. a Kähler form.
(I am not interested in a proof using Kodaira embedding thm)
One direction is easy: Let $L$ positive. Then $c_1(L)$ is a Kähler class and $deg(L)=Vol(C)>0$.
But what about the other direction? Why does $\int_C c_1(L)>0$ imply that $c_1(L)$ is positive??