Picard-Fuchs Equations for Relative Periods and Abel-Jacobi Map for Calabi-Yau Hypersurfaces

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.