Algebraic Correspondences 'Expressible' as Vector Bundles

Question

For algebraic curves $C$ over a closed field, a correspondence on $C$ is a the same thing as a divisor, and so, a line bundle on $C \times C$. Can I assume that this simplification does not extend to the case of general varieties? Does there exist a nice characterization of those correspondences 'expressible' as vector bundles?

In the case of finite fields, is there a bundle formation of the Frobenius correspondence $(v,$Fr$(v)) \in V \times V$ in terms of bundles?

I am particularly interested in the case of projective and Grassmannian spaces.

Answer 1

I am not sure what is meant by "expressible as a vector bundle". Let me just make a few elementary observations. Fulton's "Intersection theory" has a great deal of related material.

1. An algebraic correspondence is an algebraic cycle $Z$ on $X\times Y$, but there is no reason why you should get a vector bundle out of it. An algebraic cycle of codimension 1 is a Weil divisor and a line bundle is associated with a Cartier divisor (this is a typical situation when $X$ and $Y$ are curves). These two notions generalize in different ways when codimension is larger. In fact, even in the case of curves, Weil divisors and Cartier divisors on $X\times Y$ are different notions if $X$ or $Y$ (and hence $X\times Y$) is singular.

2. There is a way to associate codimension $k$ cycles (modulo some equivalence relation) to a rank $k$ vector bundle $E$. If $E$ splits into a direct sum of $k$ line bundles then we get a scheme-theoretic intersection of $k$ codim=1 cycles corresponding to the individual summands. It is an interesting and nontrivial problem to characterize complete intersections, even for a simple case of codim=2 cycles on the projective plane $\mathbb{P}^2$. (I realize that this isn't very relevant to your question about correspondences, but it gives you an idea of what happens for higher rank vector bundles.)