  We review the ghost-free four-derivative terms for chiral superfields in
$\mathcal{N}=1$ supersymmetry and supergravity. These terms induce cubic
polynomial equations of motion for the chiral auxiliary fields and correct the
scalar potential. We discuss the different solutions and argue that only one of
them is consistent with the principles of effective field theory. Special
attention is paid to the corrections along flat directions which can be
stabilized or destabilized by the higher-derivative terms. We then compute
these higher-derivative terms explicitly for the type IIB string compactified
on a Calabi-Yau orientifold with fluxes via Kaluza-Klein reducing the
$(\alpha')^3 R^4$ corrections in ten dimensions for the respective
$\mathcal{N}=1$ K\"{a}hler moduli sector. We prove that together with flux and
the known $(\alpha')^3$-corrections the higher-derivative term stabilizes all
Calabi-Yau manifolds with positive Euler number, provided the sign of the new
correction is negative.
