Chain integral solutions to tautological systems.

*(English)*Zbl 1364.14031Authors’ abstract: 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 [S. Bloch et al., J. Differ. Geom. 97, No. 1, 11–35 (2014; Zbl 1318.32027); A. Huang et al., J. Differ. Geom. 104, No. 2, 325–369 (2016; Zbl 1387.14042)]. 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.

Reviewer: Vladimir P. Kostov (Nice)

##### MSC:

14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |

14D07 | Variation of Hodge structures (algebro-geometric aspects) |

14D05 | Structure of families (Picard-Lefschetz, monodromy, etc.) |

32S35 | Mixed Hodge theory of singular varieties (complex-analytic aspects) |

34M50 | Inverse problems (Riemann-Hilbert, inverse differential Galois, etc.) for ordinary differential equations in the complex domain |

33C80 | Connections of hypergeometric functions with groups and algebras, and related topics |