Chain Integral Solutions to Tautological Systems

An Huang Harvard University Bong H. Lian Brandeis University Shing-Tung Yau Harvard University Xinwen Zhu California Institute of Technology


We give a new geometrical interpretation of the local analytic solutions to a differential system, which we call a tautological system , 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 [3, 7]. In light of this result, the mixed Hodge structure of the solution sheaf is now seen as the MHS of $H_n(X − Ya, \cup D − Y_a)$. 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.
Chain Integral Solutions, Tautological Systems
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  title={Chain Integral Solutions to Tautological Systems },
  author={An Huang, Bong H. Lian, Shing-Tung Yau, and Xinwen Zhu},
An Huang, Bong H. Lian, Shing-Tung Yau, and Xinwen Zhu. Chain Integral Solutions to Tautological Systems . 2015.
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