ar X iv : 1 50 8 . 00 40 6 v 2 [ m at h . A G ] 6 A ug 2 01 5 CHAIN INTEGRAL SOLUTIONS TO TAUTOLOGICAL SYSTEMS

We give a new geometrical interpretation of the local analytic solutions to a differential system, which we call a tautological system τ , arising from the universal family of Calabi-Yau hypersurfaces Ya in a G-variety X of dimension n. First, we construct a natural topological correspondence between relative cycles in Hn(X − Ya,∪D − Ya) bounded by the union of G-invariant divisors ∪D in X to the solution sheaf of τ , 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 Ya, 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 [3, 7]. In light of this result, the mixed Hodge structure of the solution sheaf is now seen as the MHS of Hn(X − Ya,∪D − Ya). 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.