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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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