  We give necessary conditions for the existence of perturbative heterotic and
common sector type II string warped compactifications preserving four and eight
supersymmetries to four spacetime dimensions, respectively. In particular, we
find that the only compactifications of heterotic string with the spin
connection embedded in the gauge connection and type II strings are those on
Calabi-Yau manifolds with constant dilaton. We obtain similar results for
compactifications to six and to two dimensions.
