# A remark on Beauville's splitting property

## TL;DR

This paper discusses a conjecture related to the injectivity of a subring of algebraic cycles on hyperk"ahler varieties into cohomology, providing evidence for its validity.

## Contribution

It proposes a new conjecture extending Beauville's splitting property to algebraic cycles modulo algebraic equivalence and offers supporting evidence.

## Key findings

- Evidence supporting the conjecture is presented.
- The subring containing divisors and codimension 2 cycles likely injects into cohomology.
- The conjecture extends Beauville's original splitting property.

## Abstract

Let $X$ be a hyperk\"ahler variety. Beauville has conjectured that a certain subring of the Chow ring of $X$ should inject into cohomology. This note proposes a similar conjecture for the ring of algebraic cycles on $X$ modulo algebraic equivalence: a certain subring (containing divisors and codimension $2$ cycles) should inject into cohomology. We present some evidence for this conjecture.