Hyper-K\"ahler Fourfolds Fibered by Elliptic Products

TL;DR

This paper investigates hyper-K"ahler fourfolds fibered by Abelian surfaces, specifically focusing on cases where the fibers are products of elliptic curves, extending previous classifications of Jacobian fibers.

Contribution

It provides a classification of hyper-K"ahler fourfolds with fibers that are products of elliptic curves, under mild genericity assumptions, expanding the understanding beyond Jacobian cases.

Findings

Classified hyper-K"ahler fourfolds with elliptic product fibers

Identified conditions under which fibers are elliptic products

Extended previous Jacobian fiber classifications

Abstract

Every fibration of a projective hyper-K\"ahler fourfold has fibers which are Abelian surfaces. In case the Abelian surface is a Jacobian of a genus two curve, these have been classified by Markushevich. We study those cases where the Abelian surface is a product of two elliptic curves, under some mild genericity hypotheses.