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 embedding statements.

A compact complex subvariety of ${\mathbb{C}}^n$ is a point. However, every compact real manifold of dimension $n$ can be realized as a submanifold of some ${\mathbb{R}}^{2n}$.

There are compact complex manifolds that cannot be embedded into complex projective space. An example most often quoted in textbooks is the Hopf manifold, which is not even Kahler. On the other hand, I heard that embedding into real projective space is not often considered in differential geometry.
For a closed analytic subset Z ⊂ S of a (say compact) complex manifold 
with complement U=S-Z one has additivity of the (topological) Euler characteristic:

Χ(S)=Χ(Z)+Χ(U).

This is wrong for if S and Z are topological spaces or smooth manifolds.
Indeed, take for Z a point on a circle S.
This (surprising) difference was recently pointed out to me by Manfred Lehn.

Of course there is also no additivity of Poincare-Polynomials or other 
"motivic" invariants of complex varieties.