  We study the topological conditions for general Calabi-Yaus to get a
non-supersymmetric AdS exponentially large volume minimum of the scalar
potential in flux compactifications of IIB string theory. We show that negative
Euler number and the existence of at least one blow-up mode resolving
point-like singularities are necessary and sufficient conditions for moduli
stabilisation with exponentially large volumes. We also analyse the general
effects of string loop corrections on this scenario. While the combination of
alpha' and nonperturbative corrections are sufficient to stabilise blow-up
modes and the overall volume, quantum corrections are needed to stabilise other
directions transverse to the overall volume. This allows exponentially large
volume minima to be realised for fibration Calabi-Yaus, with the various moduli
of the fibration all being stabilised at exponentially large values. String
loop corrections may also play a role in stabilising 4-cycles which support
chiral matter and cannot enter directly into the non-perturbative
superpotential. We illustrate these ideas by studying the scalar potential for
various Calabi-Yau three-folds including K3 fibrations and briefly discuss the
potential phenomenological and cosmological implications of our results.
