  It has long been known that the maximal cut of the equal-mass four-loop
banana integral is a period of a family of Calabi-Yau threefolds that depends
on the kinematic variable $z=m^2/p^2$. We show that it can also be interpreted
as a period of a family of genus-two curves. We do this by introducing a
general Calabi-Yau-to-curve correspondence, which in this case locally relates
the original period of the family of Calabi-Yau threefolds to a period of a
family of genus-two curves that varies holomorphically with the kinematic
variable $z$. In addition to working out the concrete details of this
correspondence for the equal-mass four-loop banana integral, we outline when we
expect a correspondence of this type to hold.
