Prym-period duality for semi-flat Calabi–Yau threefolds

Prove that, for a special Lagrangian fibration on a semi-flat Calabi–Yau threefold at a noncritical base point, the period lattice of the real points of the Prym variety associated with the double cover of the cubic defined by the cubic tensor and ramified over its intersection with the imaginary quadric defined by the Weil–Petersson metric is dual to the lattice of integrals of basic one-forms on the corresponding fibre against its integral first homology.

Background

The paper draws an analogy between Lagrangian fibrations on six-dimensional O’Grady manifolds and semi-flat special Lagrangian fibrations on real six-dimensional Calabi–Yau geometries. In this setting, the base carries a quadratic tensor, represented by the Weil–Petersson metric, and a cubic tensor.

The conjecture proposes a precise lattice-theoretic relationship between the Prym variety arising from the cubic and imaginary quadric on the projectivized tangent space and the integral period data of the Lagrangian fibre. This statement is presented as a further conjectural direction rather than a result proved in the paper.