Algebraic cycles on Gushel-Mukai varieties

TL;DR

This paper proves several deep conjectures for Gushel-Mukai varieties, computes their Chow groups, and explores relations between their Chow motives, advancing understanding of algebraic cycles in complex geometry.

Contribution

It establishes the generalized Hodge, Mumford-Tate, and Tate conjectures for all GM varieties and computes their integral Chow groups except for two infinite-dimensional cases.

Findings

Proved the generalized Hodge conjecture for all GM varieties.

Computed all integral Chow groups of GM varieties except two cases.

Established isomorphisms of Chow motives for generalised partners and duals.

Abstract

We study algebraic cycles on complex Gushel-Mukai (GM) varieties. We prove the generalised Hodge conjecture, the (motivated) Mumford-Tate conjecture, and the generalised Tate conjecture for all GM varieties. We compute all integral Chow groups of GM varieties, except for the only two infinite-dimensional cases (1-cycles on GM fourfolds and 2-cycles on GM sixfolds). We prove that if two GM varieties are generalised partners or generalised duals, their rational Chow motives in middle degree are isomorphic.