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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.

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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. – Jack Huizenga May 8 '12 at 6:09
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. – Jason Starr May 8 '12 at 14:03
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. – Jason Starr May 8 '12 at 14:05
I made the question more precise. – expmat May 16 '12 at 14:45
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, – Emerton May 16 '12 at 15:07
up vote 5 down vote accepted

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.

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@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? – expmat May 16 '12 at 17:16
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". – Donu Arapura May 16 '12 at 17:51
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, – Emerton May 17 '12 at 16:35
Any other references for correspondences? I don't have easy access to this paper by Kleiman. – expmat May 21 '12 at 19:26

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