  We argue that the problem of finding lower-dimensional de Sitter solutions to
the classical field equations of higher-dimensional supergravity necessarily
requires understanding the back-reaction of whatever localized objects source
the bulk fields. However, we also find that most of the details of the
back-reacted solutions are not important for determining the lower-dimensional
curvature. We find, in particular, a classically exact expression that, for a
broad class of geometries, directly relates the curvature of the
lower-dimensional geometry to asymptotic properties of various bulk fields near
the sources. Specializing to codimension-two sources, we find that the
contribution involving the asymptotic behaviour of the warp factor (which has a
definite sign for most supergravities and so is usually used to infer a
preference for anti-de Sitter geometries) is precisely canceled by the
contribution of the sources themselves (that are left out in earlier
treatments). We identify which combination of bulk fields survives this
cancelation, and so controls the sign of the lower-dimensional geometry, for
several supergravities in 6, 10 and 11 dimensions. Our results show precisely
why explicit 4D de Sitter solutions to 6D supergravity evade general no-go
theorems. As an application we show that all classical compactifications of
Type IIB supergravity (and F-theory) to 8 dimensions are 8D-flat if they
involve only the metric and the axio-dilaton sourced by codimension-two
sources, extending earlier results to include warped solutions and more general
source properties.
