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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??

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    $\begingroup$ I suspect that if you don't use the fact that $C$ is projective, you'll have to do a Hodge-theoretic argument such as on pp. 148-150 on Griffiths-Harris. Be warned that the lemma on p. 149 is wrong: You need to assume $d$-exact (which holds in this case, of course) and the proof uses both $\partial$- and $\bar\partial$-exactness. $\endgroup$ – Ted Shifrin Jul 19 '18 at 0:04

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