On the Chow ring of some Lagrangian fibrations

TL;DR

This paper investigates Beauville's splitting property conjecture for hyperk"ahler varieties with Lagrangian fibrations, analyzing specific classical examples to understand the conjectured behavior in their Chow rings.

Contribution

The paper provides a detailed study of Beauville's conjecture in the context of classical Lagrangian fibrations on hyperk"ahler varieties, offering new insights into their Chow rings.

Findings

Evidence supporting Beauville's splitting property in studied examples

Identification of specific behaviors of fibers in the Chow ring

Advancement in understanding the structure of hyperk"ahler fibrations

Abstract

Let $X$ be a hyperk\"ahler variety admitting a Lagrangian fibration. Beauville's "splitting property" conjecture predicts that fibres of the Lagrangian fibration should have a particular behaviour in the Chow ring of $X$. We study this conjectural behaviour for two very classical examples of Lagrangian fibrations.