Let $X$ be a (edit: nonsingular) projective complex algebraic curve. The genus of $X$ can be defined as the dimension of the space of holomorphic $1$-forms on $X$, which in turn can be defined either analytically or algebraically in terms of Kahler differentials. It can also be defined as the topological genus of $X$ considered as a surface, which in turn can be defined either topologically as the number of tori in a connected sum decomposition of $X$ or homologically in terms of the Betti numbers of $X$. Does anyone know of a reasonably self-contained reference where some or all of these equivalences are proven?


(There is a [related question](https://mathoverflow.net/questions/152/how-do-you-see-the-genus-of-a-curve-just-looking-at-its-function-field) about computing the genus of a curve from its function field as well as a nice [post by Danny Calegari](http://lamington.wordpress.com/2009/09/23/how-to-see-the-genus/) explaining the relationship to the Newton polygon, but I am mostly interested in the algebraic-to-topological step of going from Kahler differentials to the number of tori in a connected sum decomposition.)



Reference for equivalent definitions of the genus
For $\mathrm{dim} H^0(X, \Omega^1\_X) = \dim H^1(X, \mathbb{Q})$ see <http://en.wikipedia.org/wiki/Hodge_theory>. For $\dim H^1(X, \mathbb{Q}) =$ number of tori use induction and the Mayer-Vietoris sequence.


(And for $\mathrm{dim} H^0(X, \Omega^1\_X) = \mathrm{dim} H^1(X, \mathcal{O}\_X)$ see <http://en.wikipedia.org/wiki/Serre_duality>.)


