Denote this Riemann surface by $X$. It is equivalent to find a complex line bundle of positive degree on $X$.

We can identify the isomorphism classes of line bundles on $X$ with $H^1(X;\mathcal{C}^{\infty}_X(\mathbb{C}^*))$ via Cech data, where $\mathcal{C}^{\infty}_X(\mathbb{C}^*)$ is the sheaf of smooth $\mathbb{C}^*$-valued functions on $X$. Furthermore, the exponential sequence $$0\rightarrow \underline{\mathbb{Z}}\rightarrow\mathcal{C}^{\infty}_X(\mathbb{C})\rightarrow\mathcal{C}^{\infty}_X(\mathbb{C}^*)\rightarrow 0$$ yields a long-exact sequence in sheaf cohomology that identifies $H^1(X;\mathcal{C}^{\infty}_X(\mathbb{C}^*))$ with $H^2(X;\underline{\mathbb{Z}})\cong\mathbb{Z}$. This identification associates to a line bundle in $H^1(X;\mathcal{C}^{\infty}_X(\mathbb{C}^*))$ its degree in $\mathbb{Z}$,