In a complex analysis course I have been given the following definition:


Let $X$ be a Riemann surface, denote by $H(1,0)$ the space of all $(1,0)$-holomorphic forms on $X$ and consider the quotient vector space (over $\mathbb{C}$) of $H(1,0)$ by $$\{f \in H(1,0) \mid f = d(\phi) \text{ for some } \phi \in C(X)\}.$$ The dimension of this vector space is called the genus of the surface.


Does anyone know of any good book that deals with this?


Thank you.



Space of $(1,0)$-holomorphic forms on a Riemann surface
There is the introductory graduate-level text Riemann Surfaces by Otto Forster which approaches the subject from just the angle suggested by the definition you were given. If you read French there is the book Quelques Aspects des Surfaces de Riemann by Eric Reyssat, a gentle introduction with a broad outlook. Rather more demanding is Compact Riemann Surfaces by Raghavan Narasimhan, a modern treatment that is not overly long but covers considerable ground. Actually, there are many good books on Riemann surfaces, not all from an algebraic geometry viewpoint. If you can get your hands on them, the wonderful Columbia University notes of Lipman Bers show Riemann surfaces from a complex analysis/PDE/differential geometry angle that you should not miss. They date from 1957, so inevitably some things are not there (I don't recommend them for the specific purpose you had in mind). If your taste is towards analysis, there is also Compact Riemann Surfaces by Jürgen Jost.


I think Forster's book is my best response to your question. Or perhaps even more useful if you are in a hurry; Chapter 9 of the second edition of Complex Analysis in One Variable by Narasimhan may be all you need!


