Comments on $AdS_2$ solutions from M2-branes on complex curves and the backreacted K\"ahler geometry

We consider $AdS_2$ solutions of M-theory which are obtained by twisted compactifications of M2-branes on a complex curve. They are of a generalized class, in the sense that the non-abelian part of the connection for the holomorphic bundle over the supersymmetric cycle is nontrivial. They are solutions of $U(1)^4$ gauged supergravity in $D=4$, with magnetic flux over the curve, and then uplifted to $D=11$. We discuss the behavior of conformal fixed points as a function of the non-abelian connection. We also describe how they fit into the general description of wrapped M2-brane $AdS_2$ solutions and their higher-order generalizations, by showing that they satisfy the master equation for the eight-dimensional K\"ahler base space.