  We study the variation of relative cohomology for a pair consisting of a
smooth projective hypersurface and an algebraic subvariety in it. We construct
an inhomogeneous Picard-Fuchs equation by applying a Picard-Fuchs operator to
the holomorphic top form on a toric Calabi-Yau hypersurface, and deriving a
general formula for the $d$-exact form on one side of the equation. We also
derive a double residue formula, giving a purely algebraic way to compute the
inhomogeneous Picard-Fuchs equations for Abel-Jacobi map, which has played an
important role in recent study of D-branes [25]. Using the variation formalism,
we prove that the relative periods of toric B-branes on a toric Calabi-Yau
hypersurface satisfy the enhanced GKZ-hypergeometric system proposed in physics
literature [6], and discuss the relations between the works [25] [21] [6] in
recent study of open string mirror symmetry. We also give the general solutions
to the enhanced hypergeometric system.
