Lagrangian planes in hyperk\"ahler varieties of $K3^{[n]}$-type

TL;DR

This paper proves Jieao Song's conjecture relating the class of a Lagrangian plane to a line class in hyperk"ahler varieties of $K3^{[n]}$-type, confirming a specific geometric relationship.

Contribution

The paper provides a proof of Song's conjecture for primitive line classes, advancing understanding of Lagrangian subvarieties in hyperk"ahler geometry.

Findings

Proof of Song's conjecture for primitive line classes

Explicit formula relating Lagrangian plane class to line class

Enhanced understanding of hyperk"ahler Lagrangian subvarieties

Abstract

Jieao Song recently conjectured a formula for the class of a Lagrangian plane on a hyperk\"ahler variety of $K3^{[n]}$-type in terms of the class of a line on it. We give a proof of this conjecture if the line class is primitive.