Hi,

Given two (smooth, projective) curves $X$ and $Y$ over a field $k$, define a *correspondence* to be a line
bundle $L$ on $X\times Y$. A *trivial* correspondence is a correspondence of the form $p_1^*L_1\otimes p_2^*L_2$. Define a function $N$ on the set of correspondences by
$$N(L) = -\chi(L)+\chi_1(L)\chi_2(L),$$
where $\chi_1(L)$ (resp. $\chi_2(L)$) is the Euler characteristic of $L\mid X\times y$ (resp. $L\mid x\times Y$) for any $y\in Y$ (resp. $x\in X$) (the Euler characteristic is constant).
Then it turns out that $N$ is well defined on the set of correspondences modulo trivial corresopndences, and it is a non-degenerate quadratic form. By this I mean that $N(L)\geq 0$
and $N(L) = 0$ if and only if $L$ is a trivial correspondence ('$N$ is non-degenerate') and the form
$$
[L_1,L_2] = N(L_1\otimes L_2) - N(L_1) - N(L_2) - N(\mathcal O_{X\times Y})
$$
is bilinear ('$N$ is quadratic').

- Question: what is going on here? It just seems magic that this works. Does it work more generally? In higher dimensions? Is there a complex analytic analog of this?

Thanks!