I am following the proof to show that the complex torus is the same as the projective algebraic curve.
First we consider the complex torus minus a point, punctured torus, and show there is a biholomorphic map or holomorphic isomorphism with the affine algebraic curve in ##\mathbb{C}^2##...
For instance Alain Connes has dedicated work to Riemann's Hypothesis, who would fit the analog for this on Hodge's Conjecture? Has there been any recent progress done in the field? Since it's quite an esoteric subject of matter and with work on it being done at the best gradually to my knowledge...