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We discuss the four-dimensional N=1 effective approach in the study of warpedtype II flux compactifications with SU(3)x SU(3)-structure to AdS_4 or flatMinkowski space-time. The non-trivial warping makes it natural to use asupergravity formulation invariant under local complexified Weyltransformations. We obtain the classical superpotential from a standardargument involving domain walls and generalized calibrations and show how theresulting F-flatness and D-flatness equations exactly reproduce the fullten-dimensional supersymmetry equations. Furthermore, we consider the effect ofnon-perturbative corrections to this superpotential arising from gauginocondensation or Euclidean D-brane instantons. For the latter we derive thesupersymmetry conditions in N=1 flux vacua in full generality. We find that thenon-perturbative corrections induce a quantum deformation of the internalgeneralized geometry. Smeared instantons allow to understand KKLT-like AdSvacua from a ten-dimensional point of view. On the other hand, non-smearedinstantons in IIB warped Calabi-Yau compactifications 'destabilize' theCalabi-Yau complex structure into a genuine generalized complex one. Thisdeformation gives a geometrical explanation of the non-trivial superpotentialfor mobile D3-branes induced by the non-perturbative corrections.

From ten to four and back again: how to generalize the geometry