3
$\begingroup$

How does a correspondence on an algebraic curve $C$ induce a map on $\Omega^1_C$? Apparently it passes through the Jacobian of $C$ but I don't quite understand it.

More specifically, I was reading a paper that said roughly the following:

Let C be a curve and $\Gamma$ (with maps $\pi_i: \Gamma \rightarrow C$ for $i=1,2$) be a correspondence on $C$. The map $\pi_2$ is a double cover and $\tau: \Gamma \rightarrow \Gamma$ is a map that switches the elements of the fibers of $\pi_2$. Then the induced map on $H^0(C, \Omega^1)$ is given by: $\omega \mapsto \pi_1^* \omega + \tau^* \pi_1^* \omega$, where the differentials of $C$ are identified with the ones of $\Gamma$ that are $\tau^*$-invariant.

I don't understand why the map on $H^0(C, \Omega^1)$ is what it is claimed to be (even assuming the mentioned identification). I guess this follows from general theory of correspondences but I don't know where I would find such a statement.

$\endgroup$
7
  • $\begingroup$ Can you say a bit more about what you mean? Maps on the canonical bundle aren't particularly interesting, since they are just constants. I'm guessing you have something else in mind. $\endgroup$ May 8, 2012 at 6:09
  • 1
    $\begingroup$ Let $C_1$ and $C_2$ be smooth, projective, geometrically connected curves over a field. Let $\Gamma \subset C_1\times C_2$ be a correspondence. For every Cartier divisor $D_1$ on $C_1$, there is an associated Cartier divisor class $(\pi_2)_*(\pi_1^*D_1 \cap \Gamma)$ on $C_2$. This induces a morphism $\gamma:\text{Pic}(C_1)\to \text{Pic}(C_2)$ which may not preserve degree, yet does induce a unique morphism $\gamma^0:\text{Pic}^0(C_1) \to \text{Pic}^0(C_2)$ such that $\gamma$ becomes a morphism of homogeneous spaces under the action of $\text{Pic}^0(C_1)$. contd. $\endgroup$ May 8, 2012 at 14:03
  • $\begingroup$ contd. My understanding of the OP's question is to understand the pullback map on sheaves of relative differentials $\gamma^*:H^0(\text{Pic}^0(C_2),\Omega^1) \to H^0(\text{Pic}^0(C_1), \Omega^1)$. This is canonically equivalent to a map $H^0(C_2,\Omega^1)\to H^0(C_1,\Omega^1)$. Presumably the OP wants a more explicit description of this map, one which does not directly use that Picard of the curve. $\endgroup$ May 8, 2012 at 14:05
  • $\begingroup$ I made the question more precise. $\endgroup$
    – expmat
    May 16, 2012 at 14:45
  • 2
    $\begingroup$ A correspondence is a multi-valued map. Whenever we linearize a geometric situation (by passing to cohomology, such as $H^0(\Omega^1)$) we can make a mutlivalued map become single valued by adding up the different values. This is what is happening here (and there is no need to mention the Jacobian). Technically, if $\pi_1, \pi_2: \Gamma \to C_1,C_2$ are the two projections (where $\Gamma$ is a correspondence from $C_1$ to $C_2$, then the induced map is given by $(\pi_1)_*\pi_2^*.$ Regards, $\endgroup$
    – Emerton
    May 16, 2012 at 15:07

1 Answer 1

5
$\begingroup$

This was pretty much answered in the comments by Jason Starr and Emerton, but to elaborate a bit, the simplest type of correspondence is (the graph of) a map $f:C_1\to C_2$. In this case the action on $H^0(\Omega^1)$, or anything else, is by $f^*$. In general, given $C_1\leftarrow \Gamma\to C_2$, with maps labeled $\pi_i$, the action on differentials is by $$H^0(C_2,\Omega^1)\stackrel{\pi_2^*}{\to} H^0(\Gamma,\Omega^1)\stackrel{\pi_{1*}}{\to} H^0(C_1,\Omega^1)$$ The second map is dual to $\pi_1^*$ under Serre duality. If $\Gamma\to C_1$ is a Galois with group $G$ having order prime to the characteristic (which includes the case you care about), $\pi_{1*}$ is the projection $$H^0(\Gamma,\Omega^1)\to H^0(\Gamma,\Omega^1)^G\cong H^0(C_1,\Omega^1)$$ given by averaging over the group.

$\endgroup$
4
  • $\begingroup$ @Donu: Thanks a lot for the explanation! Now, when you say $\pi_1^*$ is the projection, do you mean $(\pi_1)_*$? Also, where can I find these statements in a way I can refer to them? Or maybe learn about them? $\endgroup$
    – expmat
    May 16, 2012 at 17:16
  • $\begingroup$ Right, I fixed that. I don't know any really easy references, but surely they exist. Perhaps some else has suggestions. I learned about correspondences from Kleiman's "Algebraic cycles and the Weil conjectures". $\endgroup$ May 16, 2012 at 17:51
  • 1
    $\begingroup$ Dear Donu, The pushforward of differentials for a finite map of curves is in some older books, maybe Lang's algebraic geometry book and/or Serre's geometric class field theory book, and this is where I first learnt it. It comes up in Serre's proof (or maybe it's originally Weil's proof?) that the sum of the residues of a differential are zero; he verifies that the statement is invariant under pushforwards, via a local computation on the target, and then reduces to the case of the projective line. Regards, $\endgroup$
    – Emerton
    May 17, 2012 at 16:35
  • $\begingroup$ Any other references for correspondences? I don't have easy access to this paper by Kleiman. $\endgroup$
    – expmat
    May 21, 2012 at 19:26

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.