Lagrangian subvarieties in the Chow ring of some hyperk\"ahler varieties

TL;DR

This paper proves a conjecture about the Chow ring of hyperk"ahler varieties, showing that certain Lagrangian subvarieties have trivial intersection with specific parts of the Chow ring, in particular cases involving Hilbert schemes with Lagrangian fibrations.

Contribution

It establishes the conjecture for Hilbert schemes with Lagrangian fibrations, demonstrating trivial intersection properties for general fibers of these fibrations.

Findings

Proves the conjecture for specific Hilbert schemes with Lagrangian fibrations

Shows Lagrangian fibers have trivial intersection with certain Chow ring components

Advances understanding of the Chow ring structure in hyperk"ahler varieties

Abstract

Let $X$ be a hyperk\"ahler variety, and let $Z\subset X$ be a Lagrangian subvariety. Conjecturally, $Z$ should have trivial intersection with certain parts of the Chow ring of $X$. We prove this conjecture for certain Hilbert schemes $X$ having a Lagrangian fibration, and $Z\subset X$ a general fibre of the Lagrangian fibration.