I am hoping this question is sensible and non-trivial. I am learning algebraic geometry at the moment, and have taken a strong liking to it. Unfortunately my complex analysis is weaker and I only know it at an undergraduate level. I am trying to transfer some of what I know in algebraic geometry to the language of complex analysis, particularly complex manifolds.
My main question is, what are the benefits and drawbacks of defining the sheaf on a complex manifold (for the time being say a Riemann surface, or even just the Riemann sphere) in terms of holomorphic functions as opposed to meromorphic functions? From what I have gathered, meromorphic functions align better with the theory of discrete valuation rings on algebraic curves, since this provides a framework for studying poles. However it seems that holomorphic functions are taken to be the standard structure sheaf. What difference does this make, and why do you choose one over the other in certain situations? Does it make any difference to the sheaf cohomology? Does it make a difference if the surface is compact or not?
Again, forgive me if this question is either trivial or not particularly meaningful, but I feel like it would massively boost the speed I can learn complex geometry if I can frame it in the language of ringed spaces and algebraic geometry.
Any help is appreciated, or even some introductory notes that you think would help someone coming from my perspective.