Beauville’s splitting property conjecture for Chow rings

Establish that, for the special varieties envisioned by Beauville, the Chow ring admits a multiplicative splitting, thereby realizing the proposed splitting property conjecture.

Background

The paper motivates multiplicative Chow–Künneth decompositions through Beauville’s proposed splitting property for Chow rings. The conjecture seeks a multiplicative decomposition of the Chow ring for appropriate classes of special varieties, analogous to the structures known for K3 surfaces and abelian varieties.

The authors explain that Shen–Vial’s notion of a multiplicative Chow–Künneth decomposition provides a concrete framework in which Beauville’s otherwise elusive splitting property can be formulated and studied. The cited passage presents the conjecture as an unresolved general expectation, although the paper establishes specific positive and negative results for Fano threefolds.

References

Motivated by this particular behaviour of K3 surfaces and abelian varieties, Beauville has conjectured that for certain special varieties, the Chow ring should admit a multiplicative splitting.

— Algebraic cycles and Fano threefolds of genus 7  (2608.12950 - Laterveer, 13 Aug 2026) in Section 1, Introduction

It is conjectured that for any $X$ with an MCK decomposition, one has

\Ai_{(j)}(X)\stackrel{??}{=}0\ \ \ \hbox{for}\ j<0\ ,\ \ \Ai_{(0)}(X)\cap \Ai_{hom}(X)\stackrel{??}{=}0\ ;

this is related to Murre's conjectures B and D, that have been formulated for any CK decomposition , .

— Algebraic cycles and Fano threefolds of genus 7  (2608.12950 - Laterveer, 13 Aug 2026) in Section 2, Subsection “MCK decomposition,” Remark following the definition of MCK decomposition