Transplantation and isogeny of intermediate Jacobians of compact K\"ahler manifolds

TL;DR

The paper introduces a novel method to construct compact K"ahler manifolds with isogenous intermediate Jacobians, extending Sunada's technique to complex geometry and providing explicit examples.

Contribution

It develops a general transplantation method for creating K"ahler manifolds with isogenous intermediate Jacobians, linking spectral geometry with complex algebraic geometry.

Findings

Constructed examples of K"ahler manifolds with isogenous intermediate Jacobians.

Extended Sunada's spectral technique to complex geometric structures.

Produced manifolds with isogenous Lazzeri Jacobians.

Abstract

We give a general method for constructing compact K\"ahler 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.