I believe that a typical nilmanifold, such as upper 3x3 real matrices modulo upper 3x3 integer matrices, will be compact but have a noncompact isometry group.

They are all either reversed examples of pulling back a universal object along a map from B, or else fail to be an example of this in an interesting (interesting-to-me at least) way.

I am also wrong about torsion, the example in my previous comment is bogus. I've asked another question about this.

I am wrong about X having to be compact, since U and X will have a common compactification. Torsion Chern classes still give obstructions: e.g. if X is the total space of O(n), n < -1, on P^1 and U is the complement of the zero section, there are line bundles on U that do not extend across X or a compactification of X.

4. is nice! It requires X smooth and compact, so Chern classes can still give obstructions e.g. to extending vector bundles across the special fiber of a degeneration.

A basic obstruction is topological/cohomological: if the Chern classes of F are not in the image of $H^*(X) \to H^*(U)$, then F will not extend. Sasha gives an example where this obstruction vanishes and F still does not extend.

Here are two problematic singular examples: the normalization of a cusp like y^2 = x^3 is not flat over the cusp, but the underlying map on spaces is a homeomorphism. And any reasonable family connecting the union of the x- y- and z-axes in C^3 to the union of three different lines in C^2 will not be flat, but will be topologically of the form X x C --> C.

Two proto-cohomological ways in which a smooth curve X/k resembles a topological surface: if you remove finitely many k-points from X, X remains connected. And there exists a finite map Y --> X from a smooth Y which is ramified over and unramified away from some points of X. For a manifold, the first property implies that a point has codimension at least 2 and the second property implies that a point has codimension at most 2.

I don't think I've understood it yet, but this is a cool interpretation. Tell me more about your catch: presumably to talk about whether or not a harmonic differential form is integral you are using the identification $\mathcal{H}^{0,2} + \mathcal{H}^{2,0} = H^{0,2} + H^{2,0}$. Isn't the right hand side, as a subspace of $H^2(X;C)$, independent of the Kahler form and even the Kahler class chosen?

$V$ is the space of linear functions on $V^*$, and Sym($V$) is the space of polynomial functions on $V^*$, so that Spec(Sym($V$)) is naturally identified with the dual vector space (or dual vector bundle) to $V$. This explains your tangent/cotangent concern. After replacing $\Omega_{Y/X}$ by its dual, the answer to your auxiliary question 1. is "yes," and to answer question 2. using this description note that the S-points of GL_n,Y are the S-automorphisms of O_S^{\oplus n}.

The profinite dihedral group (the extension of Z/2 by Z-hat) has finite abelianization but infinitely many 2-dimensional representations.

Our intuition is wrong! The alternating sum of the eigenspaces does not depend on the eigenvalue, so that is some kind of relation.