The Atiyah-Patodi-Singer index theorem, applies to differential operators $D:\Gamma(E)\to \Gamma(F)$ over a manifold $X$ with a collar on the boundary isometric to $[0,1]\times Y$. The operator must be written, over the collar as
$$D = \sigma(\partial_t + A)$$ where $t$ is the coordinate normal to the boundary and $A$ is a first order self-adjoint operator over $Y$ (so $A$ does not depend on $t$) and $\sigma$ is a bundle isomorphism.
I would like to apply this to the Cauchy-Riemann operator $\bar \partial$ over a punctured Riemann surface. This seems to be possible as it is mentioned in the first paper by Atiyah, Patodi and Singer (pg 62).
However, if we consider a neighbouhood of the puncture (this is diffeomorphic to $\mathbb{S}^1\times [0,1]$) then in polar coordinates the operator looks like $\partial_r - \frac 1 rJ \partial_\theta$ where $J$ is the almost complex structure, $r$ the radial coordinate and $\theta$ the angular coordinate (see also Cauchy-Riemann equations in polar form.). In particular the operator $A$ in this case depends on the radius (and conjugating by isomorphisms of the bundle does not help).
How can I put $\bar \partial$ in a suitable form to apply the APS theorem?
