At the leading order, the low-energy effective field equations of string theory admit solutions in the form of products of Minkowski spacetime with a Ricci-flat Calabi-Yau space. The equations of motion receive corrections at higher orders in α', which imply that the Ricci-flat Calabi-Yau space is modified. In an appropriate choice of scheme, the corrected Calabi-Yau space remains of Kähler structure, but is no longer Ricci-flat. We discuss the nature of these corrections at order α'3, and consider the deformations of the known cohomogeneity-one non-compact Kähler metrics in six and eight dimensions. We do this by deriving the first-order equations associated with the modified Killing-spinor conditions, and we thereby obtain the modified supersymmetric solutions. We also give a detailed discussion of the boundary terms for the Euler complex in six and eight dimensions, and apply the results to the cohomogeneity-one examples.