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.

Comment: 23 pages, final version, in special volume in honour of C. Voisin

• 14C15 - (Equivariant) Chow groups and rings; motives
• 14C25 - Algebraic cycles
• 14C30 - Transcendental methods, Hodge theory (algebro-geometric aspects)
• 14F08 - Derived categories of sheaves, dg categories, and related constructions in algebraic geometry
• 14J45 - Fano varieties