  We give a new geometrical interpretation of the local analytic solutions to a
differential system, which we call a tautological system $\tau$, arising from
the universal family of Calabi-Yau hypersurfaces $Y_a$ in a $G$-variety $X$ of
dimension $n$. First, we construct a natural topological correspondence between
relative cycles in $H_n(X-Y_a,\cup D-Y_a)$ bounded by the union of
$G$-invariant divisors $\cup D$ in $X$ to the solution sheaf of $\tau$, in the
form of chain integrals. Applying this to a toric variety with torus action, we
show that in addition to the period integrals over cycles in $Y_a$, the new
chain integrals generate the full solution sheaf of a GKZ system. This extends
an earlier result for hypersurfaces in a projective homogeneous variety,
whereby the chains are cycles. In light of this result, the mixed Hodge
structure of the solution sheaf is now seen as the MHS of $H_n(X-Y_a,\cup
D-Y_a)$. In addition, we generalize the result on chain integral solutions to
the case of general type hypersurfaces. This chain integral correspondence can
also be seen as the Riemann-Hilbert correspondence in one homological degree.
Finally, we consider interesting cases in which the chain integral
correspondence possibly fails to be bijective.
