# zbMATH — the first resource for mathematics

Chain integral solutions to tautological systems. (English) Zbl 1364.14031
Authors’ 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.

##### 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
Full Text: