We develop a perturbative expansion scheme for solving general boundary value problems in a broad class of type IIB flux compactifications. The background solution is any conformally Calabi-Yau compactification with imaginary self-dual (ISD) fluxes. Upon expanding in small deviations from the ISD solution, the equations of motion simplify dramatically: we find a simple basis in which the n-th order equations take a triangular form. This structure implies that the system can be solved iteratively whenever the individual, uncoupled equations can be solved. We go on to demonstrate the solution of the system for a general warped Calabi-Yau cone: we present an algorithm that yields an explicit Green's function solution for all the supergravity fields, to any desired order, in terms of the harmonic functions on the base of the cone. Our results provide a systematic procedure for obtaining the corrections to a warped throat geometry induced by attachment to a compact bulk. We also present a simple method for determining the sizes of physical effects mediated through warped geometries.

52 pages