  The study of the geometry of Calabi-Yau fourfolds is relevant for
compactifications of string theory, M-theory, and F-theory to various
dimensions. This work introduces the mathematical machinery to derive the
complete moduli dependence of the periods of non-trivial three-forms for
fourfolds realized as hypersurfaces in toric ambient spaces. It sets the stage
to determine Picard-Fuchs-type differential equations and integral expressions
for these forms. The key tool is the observation that non-trivial three-forms
on hypersurfaces in toric ambient spaces always stem from divisors that are
build out of toric resolution trees fibered over Riemann surfaces. The
three-form periods are then non-trivially related to the one-form periods of
these surfaces. In general, the three-form periods are known to vary
holomorphically over the complex structure moduli space and play an important
role in the effective actions arising in fourfold compactifications. We discuss
two explicit example fourfolds for F-theory compactifications in which the
three-form periods determine axion decay constants.
