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Let $\pi: X \to Y$ be a finite morphism between smooth projective curves over the complex numbers. I would like to known:

(1) what the Gauss-Manin connection with respect to $\pi$ (that is, the connexion corresponding to the local system $\pi_\ast\mathbb{C}$ on $Y$ minus the ramification points) looks like

(2) what kind information does the Grothendieck-Riemann-Roch theorem provide when applied to $\pi$

Thanks!

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I'll take a stab at these:

(1) Well, the Gauss-Manin connection is flat, and all flat connections look alike in local holomorphic trivializations. A better question is what the parallel sections of the local system look like. Here a simple example might be useful.

Let $\pi : \mathbb C \to \mathbb C$ be $z \mapsto z^2$. This is a finite morphism, of the admittedly non-projective affine plane to itself, but it can be extended to a finite morphism on the projective line that is ramified at $0$ and $\infty$ only. The only nontrivial local system associated to this morphism, outside of the ramification points, is a copy if two disjoint $\mathbb C$, one for each point in the preimage of a given point. A parallel section of the associated bundle over $U \subset \mathbb C$ then corresponds to the choice of a square root of $z$ over $U$, and if $U$ is connected this choice of square root does not "jump" between branches, which would correspond to jumping from one point in a preimage of $\pi$ to another.

The case of a general finite morphism should maybe be thought of as similar to this one; parallel sections of the vector bundle associated to the local system correspond to picking a branch of local solutions $x$ of $\pi(x) = y$ when $y$ varies on $Y$.

(2) I haven't worked out the details, but I'm willing to bet good money that we get an extreme overkill proof of the Riemann-Hurwitz formula by applying Grothendieck-Riemann-Roch to the finite morphism $\pi : X \to Y$.

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NB: There's something subtle going on due to the fact that the fibers of $\pi$ are not connected; in particular the "vector bundle" associated to the local system is actually a disjoint union of line bundles and not an honest vector bundle. Thus the space of sections of this "bundle" is a disjoint union of vector spaces, and not itself a vector space. –  Gunnar Magnusson Nov 23 '12 at 11:02
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