Hyperkahler manifolds of Jacobian type

TL;DR

This paper introduces the concept of hyperk"ahler manifolds of Jacobian type, explores their properties, and provides results on their non-existence in certain cases, along with implications for cubic fourfolds and the Hodge conjecture.

Contribution

It defines hyperk"ahler manifolds of Jacobian type, proves non-existence results for general polarized hyperk"ahler fourfolds of K3^{[2]}-type, and relates these to cubic fourfolds and the Hodge conjecture.

Findings

Very general polarized hyperk"ahler fourfolds of K3^{[2]}-type are not of Jacobian type.

Potential link between rational cubic fourfolds and Jacobian type of their varieties of lines.

Proved the Hodge conjecture in degree 4 for generic K3^{[2]}-type hyperk"ahler manifolds.

Abstract

In this paper we define the notion of a hyperk\"ahler manifold (potentially) of Jacobian type. If we view hyperk\"ahler manifolds as "abelian varieties", then those of Jacobian type should be viewed as "Jacobian varieties". Under a minor assumption on the polarization, we show that a very general polarized hyperk\"ahler fourfold $F$ of $K3^{[2]}$-type is not of Jacobian type. As a potential application, we conjecture that if a cubic fourfold is rational then its variety of lines is of Jacobian type. Under some technical assumption, it is proved that the variety of lines on a rational cubic fourfold is potentially of Jacobian type. We also prove the Hodge conjecture in degree 4 for a generic $F$ of $K3^{[2]}$-type.