Given an element $\omega \in H^{(1,1)}(X,\mathbb{C})$ how to construct a line bundle with chern class $\omega$?

Question

Let $X$ be a Kahler complex variety of dimension $1$. Let $\omega\in H^{(1,1)}_{dR}(X,\mathbb{C})$ such that $\int_{X}\omega = 0$. Can we find a line bundle $L$ over $X$ such that $c_1(L) = \omega$? If the answer is affirmative, how can we do it?.