We consider flux compactification of type IIB string theory as the
orientifold limit of an F-theory on a Calabi–Yau 4-fold. We show that
when supersymmetry is dominantly broken by the axion–dilaton and the
contributions of the F-terms associated with complex structure moduli
are small, the Hessian of the flux potential always has tachyonic modes
for de Sitter vacua. This implies that there exist no meta-stable de Sitter
vacua in this limit. Moreover, we find that the stability requirement
imposes a relation between the values of cosmological constant and the
scale of supersymmetry-breaking for non-supersymmetric anti-de Sitter
vacua in this limit. The proof is general and does not rely on the details
of the geometry of the compact Calabi–Yau internal space. We finally
analyze the consequences of these constraints on the statistics of metastable
de Sitter vacua and address some other related issues.