A moduli theoretic approach to Lagrangian subvarieties of hyperk\"ahler varieties: Examples

TL;DR

This paper introduces conjectures on a moduli theoretic method to construct Lagrangian subvarieties in hyperk"ahler varieties, verifies them in specific cases, and confirms related conjectures about Lagrangian coverings.

Contribution

It proposes new conjectures linking moduli spaces to Lagrangian subvarieties and verifies these in classical examples, advancing understanding of hyperk"ahler geometry.

Findings

Verification of conjectures in several cases

Recovery of classical Lagrangian examples

Confirmation of O'Grady's conjecture in specific instances

Abstract

We propose two conjectures on a moduli theoretic approach to constructing Lagrangian subvarieties of hyperk\"ahler varieties arising from the Kuznetsov components of cubic fourfolds or Gushel--Mukai fourfolds. Then we verify the conjectures in several cases, recovering classical examples. As a corollary, we confirm a conjecture of O'Grady in several instances on the existence of Lagrangian covering families for hyperk\"ahler varieties.