As we all know, a smooth 1-dimension projective variety $V$ has a important invariants up to isomorphism, which is called genus and defined by divisors. Further, if $V$ is defined over $\mathbb{C}$, then $V$ is a compact Riemann surface. On the other hand, every compact Riemann surface is orientable and is a closed surface, i.e., 2-dimension connect smooth real manifold, and thus a Riemann surface has a "genus"(in Classification theorem of closed surfaces). Is the genus of a projective variety over $\mathbb{C}$ equal to its genus as a Riemann surface (a closed surface).
Moreover, is every compact Riemann surface 1-1 corresponding to a smooth 1-dimension projective variety $V$ defined over $\mathbb{C}$? On many book, there are some images show that they seem to be equivalent. But I can not prove it. Thanks for reading and I would appreciate it if you could solve my problem.
