I was wondering if there is an easy way to compute the topological index (in the sense of Atiyah-Singer) of a (Fredholm) elliptic operator on a Riemann surface. I'm aware the Todd class is very simple in dimension 1, but I'm not sure how to compute the Chern class of such an operator.
Is there a reasonable way to do this in general? If not, I am particularly interested in working it out for the Laplacian operator, is there a reference where that is done?