We give a general method for constructing compact Kähler manifolds $X_1$ and $X_2$ whose intermediate Jacobians $J^k(X_1)$ and $J^k(X_2)$ are isogenous for each $k$, and we exhibit some examples. The method is based upon the algebraic transplantation formalism arising from Sunada's technique for constructing pairs of compact Riemannian manifolds whose Laplace spectra are the same. We also show that the method produces compact Riemannian manifolds whose Lazzeri Jacobians are isogenous.
"Sunada transplantation and isogeny of intermediate Jacobians of compact Kähler manifolds." Tohoku Math. J. (2) 72 (1) 127 - 147, 2020. https://doi.org/10.2748/tmj/1585101624