Question: I have the following setup:

There is a collection of items I and a collection of partial rankings V. That is, an element of V is a total ordering on a subset of I. There is no expectation of consistency among the elements of V: it may be that x < y for one element and y < x for another. 

I would like to assign a score $s : I \to \mathbb{R}$ which in some sense captures these rankings. That is, I would like s(x) < s(y) to mean "x tends to be less than y for elements of V which have both in their domain". I'm not sure of what a good way to do this is.

[Arrow's impossibility theorem](http://en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem) puts some constraints on what can be achieved here, because given a set of votes and a scoring function like this we could use the scoring function to define a total order on the items, which is then constrained by the theorem.

I suppose I'm really looking for references rather than an answer to this question (although both would be appreciated): I'm sure there's a body of theory around this, but I have no idea what it is like or what it's called, so I'm at a bit of a loss as to where to start looking for a solution.

Answer: 
Some of these properties are local, and distinguish analytic and algebraic functions, from smooth functions.

Others are global and distinguish compact manifolds from projective manifolds by their difference in containment of many subvarieties.

some are as simple as contrasting the dimension, and homology of say projective Real space from that of projective Complex space, as noted.

As a deep contrast between smooth and analytic structure I like the answer pointing out that analytic riemann surfaces can have many non isomorphic structures on the same smooth manifold. this is interesting already for genus one manifolds. and it is not trivial to show that the sphere has only one complex analytic structure.

that might be a fun challenge to a class, to show two Riemann surfaces both diffeomorphic to the 2-sphere, are holomorphically isomorphic.

You might also be interested in some of the articles by Kolla'r on the Nash conjecture contrasting real varieties and real manifolds. such as "What are the simplest varieties?", Bulletin, vol 38. I like the pair of theorems 54, 51, subtitled respectively: "the Nash conjecture is true in dim 3", and "The Nash conjecture is false in dim 3".
I think there's some big difference concerning the metric approach too.

In fact, the Gram-Schmidt process (which is real analytic) enables us -in real differential geometry- to find some local orthonormal frames (for any hermitian bundle, and in particular for the tangent bundle), whereas in the holomorphic case, very subtle differences may occur there.

For example, in the Kähler case, we can find "orthonormal" frames for the tangent bundle at order 2, which is the key for the Kähler identities, leading to fundamental results like the equality of all Laplacians and thus the Hodge decomposition theorem in the compact case.