On the variety of triangles for a hyper-Kaehler fourfold constructed by Debarre and Voisin

TL;DR

This paper explores the geometric properties of certain hyper-Kaehler fourfolds, focusing on the analog of triangles within these varieties and their Lagrangian subvariety structures.

Contribution

It introduces a notion of 'triangle' for hyper-Kaehler fourfolds constructed by Debarre and Voisin and proves the Lagrangian property of the associated six-dimensional variety.

Findings

The six-dimensional variety of 'triangles' is Lagrangian.

Analogies between different hyper-Kaehler fourfolds are established.

Structural properties of 'triangles' in these varieties are characterized.

Abstract

We study the similarities between the Fano varieties of lines on a cubic fourfold, a hyper-Kaehler fourfold studied by Beauville and Donagi, and the hyper-Kaehler fourfold constructed by Debarre and Voisin. We exhibit an analog of the notion of "triangle" for these varieties and prove that the 6-dimensional variety of "triangles" is a Lagrangian subvariety in the cube of the constructed hyper-Kaehler fourfold.