I am attempting to review a certain proof of the Riemann-Roch theorem, a key step of which is "elliptic bootstrapping." In particular, let $X$ be a compact Riemann surface, and $E$ be a holomorphic line bundle. We have the Chern connection between the Sobolev spaces $$\overline{\partial}:L_s(X,E)\rightarrow L_{s-1}(X,E\otimes\Lambda^{0,1}).$$ My plan of action for finding the parametrix is to choose a partition of unity subordinate to the coordinate charts, then map each chart inside a standard torus, where I can easily find a parametrix $P_{torus}$ to $\overline{\partial}$ using Fourier series. Locally, this solves the equation $$\overline{\partial}P_{torus} = I-\pi_0$$ where $\pi_0$ is projection onto the constant Fourier coefficient. This solution, though, may not be (probably won't be) supported in the P.O.U. So to pull it back to a function on $X$, I need to multiply $P_{torus}\phi$ by another bump function, supported within the chart and equal to 1 on the support of the P.O.U. Then, I can sum all the local solutions (hopefully) to get a parametrix for $\overline{\partial}$ on $X$.

Is this plan fruitful? Is the resulting operator $\overline{\partial}P-I$ compact (and regularity increasing) from $$L_{s-1}(X,E\otimes\Lambda^{0,1})\rightarrow L_{s}(X,E\otimes\Lambda^{0,1})?$$ If so, why? If not, what am I doing wrong?

P.S. Tom Mrowka presented this proof in a class, and I should have taken better notes.