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At the leading order, the low-energy effective field equations in string
theory admit solutions of the form of products of Minkowski spacetime and a
Ricci-flat Calabi-Yau space. The equations of motion receive corrections at
higher orders in \alpha', which imply that the Ricci-flat Calabi-Yau space is
modified. In an appropriate choice of scheme, the Calabi-Yau space remains
Kahler, but is no longer Ricci-flat. We discuss the nature of these
corrections at order {\alpha'}^3, and consider the deformations of all the
known cohomogeneity one non-compact Kahler 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 all the cohomogeneity one examples. Additional material will be
presented concerning the case of holonomy G_2.