I am looking for a differential geometric version of the proof of the Riemann--Roch theorem 
for Riemann surfaces, that is, $1$-dimensional compact complex manifolds. The proofs one usually finds are given in algebraic geometric terms, and can be seen as special cases of the sheaf theoretic approach of the general Hirzebruch--Riemann--Roch theorem. What I would like is a 
specialization of the proof of the Atiyah--Singer theorem to the $1$-dimensional case. 



A differential geometric proof of the Riemann--Roch theorem for lines
The details of the heat kernel proof for Atiyah-Singer index theorem become much simpler for Riemann surfaces (using a compatible Riemannian metric): a not too difficult computation shows that the first non-trivial term of the asymptotic expansion of the heat kernel along the diagonal is given by a certain combination of the scalar curvature and the curvature of the line bundle. In dimension two, this is the only relevant term for computing the index. Details can be found in Roe's book on Elliptic operators, and most likely also in Gilkey's book Invariance theory.... . You should also have a look at the answers to the question [What do heat kernels have to do with the Riemann-Roch theorem and the Gauss-Bonnet theorem?](https://mathoverflow.net/questions/14714/what-do-heat-kernels-have-to-do-with-the-riemann-roch-theorem-and-the-gauss-bonn)


