I am trying to understand that all compact genus 1 Riemann surfaces are biholomorphic to a smooth cubic curve. ( assuming that $\dim H^{1,0} = \dim H^{0,1} = \frac{1}{2} \dim H^1$ )
I think I understand how it can be shown that they are all biholomorphic to a torus $\mathbb{C}/\Lambda$ for some lattice $\Lambda$, but I am stuck to show it for the curve.
I would really appreciate a hint. Am I missing something that makes it obvious that the smooth cubic curve would be biholomorphic to the torus ?