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.
Thanks
Best Answer
If you are a complex analyst, you cannot not like holomorphic functions, in many ways they are preferable to meromorphic ones. For instance, your first preference is to solve, say, differential equations so that solutions exist everywhere (in the domain where the equation is defined and is suitably regular), rather than on an open subset. You also may want to have finite dimensionality of the space of solutions (and having estimates on solutions in terms of equations themselves or other data). Your problem, however is that compact complex manifolds lack any nonconstant holomorphic functions. As the result, you compromise and work with sheaves which may come in several different forms. One of these is the sheaf of sections of, say, a line bundle (or, more generally, a vector bundle) given by a divisor, or in form of holomorphic tensors. The latter are no longer functions and thus could exist everywhere on your compact manifold. Of course, in the process you may want to work with meromorphic functions (which are allowed to blow up at the given divisor $D$), but if you do not impose any restrictions along $D$, then you loose finite-dimensionality of the space of solutions, integrability, etc. Another thing which may happen is that solutions of your equations are multi-valued. This is not good for a variety of reasons, so you try to make them single-valued by regarding them as sections of a certain sheaf and then extending holomorphically over the branching divisor. The fact that this is (sometimes) possible is due to the fact that you are working with either holomorphic equations or, at worst, equations which are meromorphic but have controlled singularities along $D$.
The bottom line: Working with holomorphic sheaves is not that different (or, frequently, is the same) than working with meromorphic sheaves where singularities are tightly controlled (where they can occur and what type is allowed). Loosening this control may (and, frequently, does) lead to undesirable results.