The Backreacted K\"ahler Geometry of Wrapped Branes

For supersymmetric solutions of D3(M2) branes with AdS3(AdS2) factor, it is known that the internal space is expressible as U(1) fibration over K\"ahler space which satisfies a specific partial differential equation involving the Ricci tensor. In this paper we study the wrapped brane solutions of D3 and M2-branes which were originally constructed using gauged supergravity and uplifted to D=10 and D=11. We rewrite the solutions in canonical form, identify the backreacted K\"ahler geometry, and present a class of solutions which satisfy the Killing spinor equation.